![10 Principles of Real Analysis
consisting of all real numbers between a and b (excluding a and b) is called an open interval and is
denoted by ]a, b[.
Closed Interval. The set
{ : }
≤ ≤
x a x b
consisting of a, b and all real numbers lying between a and b is called a closed interval and is denoted
by [a, b].
Semi-closed or Semi-open Intervals
The intervals ] , ] { : }
= < ≤
a b x a x b and,
[ , [ { : }
= ≤ <
a b x a x b .
are semi-closed or semi-open. The former is open at a and closed at b while the latter is closed at a and
open at b.
1.3 BOUNDED AND UNBOUNDED SETS: SUPREMUM, INFIMUM
A set S of real numbers is said to be bounded above if ∃ a real number K such that every member of S
is less than or equal to K,
i.e., ,
x K x S
≤ ∀ ∈ .
The number K is called an upper bound of S. If no such number K exists, the set is said to be
unbounded above or not bounded above.
The set S is said to be bounded below if ∃ a real number k such that every member of S is greater
than or equal to k,
i.e., ,
k x x S
≤ ∀ ∈ .
The number k is called a lower bound of S. If no such number k exists, the set is said to be unbounded
below or not bounded below.
A set is said to be bounded if it is bounded above as well as below.
It may be seen that if a set has one upper bound, it has an infinite number of upper bounds. For if K
is an upper bound of a set S then every number greater than K is also an upper bound of S. Thus every set
S bounded above determines an infinite set—the set of its upper bounds. This set of upper bounds is
bounded below in as much as every member of S is a lower bound thereof. Similarly, a set S bounded
below determines an infinite set of its lower bounds, which is bounded above by the members of S.
A member G of a set S is called the greatest member of S if every member of S is less than or equal
to G, i.e.,
(i) G S
∈
(ii) ,
x G
≤ x S
∀ ∈
Similarly, a member g of the set is its smallest (or the least) member if every member of the set is
greater than or equal to g.
Clearly, a set may or may not have the greatest or the least member but an upper (lower) bound of
the set, if it is a member of the set, is its greatest (least) member. A finite set always has the greatest as
well as the smallest member.
If the set of all upper bounds of a set S has the smallest member, say M, then M is called the least
upper bound (l.u.b.) or the supremum of S.](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-25-320.jpg)
![12 Principles of Real Analysis
7. The set 4
1
:
S n
n
= ∈
N is bounded. The supremum 1 belongs to S4 while infimum 0 does not.
8. Each of the following intervals is bounded:
[a, b], ]a, b], [a, b[, ]a, b[.
Example 1.3. Prove that the greatest member of a set, if it exists, is the supremum (l.u.b.) of the set.
Solution. Let G be the greatest member of the set S.
Clearly,
,
x G x S
≤ ∀ ∈
so that G is an upper bound of S.
Again no number less than G can be an upper bound of S for if y be any number less than G, there
exists at least one member g of S which is greater than y.
Thus, G is the least of all the upper bounds of S, i.e., G is the supremum of S.
EXERCISE
1. Give examples of sets which are
(i) Bounded
(ii) Not bounded
(iii) Bounded below but not bounded above
(iv) Bounded above but not bounded below.
2. Find the infimum and the supremum of the following sets. Which of these belongs to the set?
(i) [1, 3, 5, 7, 9] (ii)
1 1 1
1, , , , ...
2 3 4
− − − −
(iii)
1
; n
n
∈
N (iv)
( 1)
;
n
n
n
−
∈
N
(v)
3 4 5 1
2, , , , ..., , ...
2 3 4
n
n
+
− − − − −
(vi)
( 1)
1 ;
n
n
n
−
+ ∈
N
(vii) ]a, b[ (viii) [a, b[.
3. Which of the sets in question 2 are bounded?
4. Find the smallest and the greatest members (if they exist) for sets in question 2.
5. Show that the greatest (or the smallest) member of a set, in case it exists, is unique.
6. Show that the smallest member of a set, if it exists, is the infimum of the set.
7. Is the converse of the solved example 1.3, true?
ANSWERS
2. (i) 1, 9; both (ii) –1, 0; infimum
(iii) 0, 1; supremum (iv)
1
1, ; both
2
−
(v) –2, –1; infimum (vi)
3
0, ; both
2
(vii) a, b; none (viii) a, b; infimum.](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-27-320.jpg)
![Real Numbers 21
Second Method: When 0.
x y
+ ≥
x y x y
+ = +
x y
≤ + and
x x y y
≤ ≤
3
When x + y < 0,
( ) ( ) ( )
x y x y x y
+ = − + = − + −
x y
≤ − + − and
x x y y
− ≤ − − ≤ −
3
But ,
x x y y
− = − =
Thus in either case,
.
x y x y
+ ≤ +
(ii) First Method:
2 2 2 2
( ) 2
x y x y x y xy
− = − = + −
2 2
2
x y x y
≥ + − ⋅
( )
xy xy x y
− ≥− = − ⋅
3
( )
2
2
x y x y
= − = −
Since, | | and | | | |
x y x y
− − are both non-negative, therefore taking the positive square
root of both sides, we have
.
x y x y
− ≥ −
Second Method:
Now ( )
x x y y x y y
= − + ≤ − + [by part (i)]
∴ x y x y
− ≥ − ...(1)
Again
( ) .
y y x x y x x
= − + ≤ − +
∴ ( )
y x y x x y
− ≥ − = − −
But y x x y
− = −
∴ ( )
x y x y
− ≥ − − ...(2)
From (1) and (2),
( )
{ }
max ,
x y x y x y
− ≥ − − −
x y
= −
Hence, x y x y
− ≥ −](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-36-320.jpg)
![CHAPTER 2
Limit Points:
Open and Closed Sets
2.1 INTRODUCTION
In this chapter, we shall study the concept of neighbourhood of a point, open and closed sets, and limit
points of a set of real numbers and the Bolzano-Weierstrass theorem, which is one of the most fundamental
theorems of Real Analysis and lays down a sufficient condition for the existence of limit points of a set.
We shall be dealing only with real numbers and sets of real numbers unless otherwise stated.
2.1.1 Neighbourhood of a Point
A set N ⊂ R is called the neighbourhood of a point a, if there exists an open interval I containing a and
contained in N,
i.e., a I N
∈ ⊂
It follows from the definition that an open interval is a neighbourhood of each of its points. Though
open intervals containing the point are not the only neighbourhoods of the point but they prove quite
adequate for a discussion like ours and are more expressive of the idea of neighbourhoods as understood
in ordinary language. We shall, therefore, whenever convenient, take the open interval ] – , [
a a
δ δ
+
where 0
δ > as a neighbourhood of the point a.
Deleted Neighbourhoods
The set { : 0 | | },
x x a δ
< − <
i.e., an open interval ] – , [
a a
δ δ
+ from which the number a itself has been excluded or deleted is
called a deleted neighbourhood of a.
Note. For the sake of brevity, we shall write neighbourhood as ‘nbd’.
ILLUSTRATIONS
1. The set R of real numbers is the neighbourhood of each of its points.
2. The set Q of rationals is not the nbd of any of its points.
3. The open interval ]a, b[ is nbd of each of its points.
4. The closed interval [a, b] is the nbd of each point of ]a, b[ but is not a nbd of the end points a
and b.](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-38-320.jpg)
![24 Principles of Real Analysis
5. The null set φ is a nbd of each of its points in the sense that there is no point in φ of which it
is not a nbd.
Example 2.1. A non-empty finite set is not a nbd of any point.
Solution. A set can be a nbd of a point if it contains an open interval containing the point. Since an
interval necessarily contains an infinite number of points, therefore, in order that a set be a nbd of a point
it must necessarily contain an infinity of points. Thus a finite set cannot be a nbd of any point.
Example 2.2. Superset of a nbd of a point x is also a nbd of x. i.e., if N is a nbd of a point x and
M N
⊃ , then M is also a nbd of x.
Example 2.3. Union (finite or arbitrary) of nbds of a point x is again a nbd of x.
Example 2.4. If M and N are nbds of a point x, then show that M N
∩ is also a nbd of x.
Solution. Since M, N are nbds of x, ∃ open intervals enclosing the points x such that
] [ ] [
1 1 2 2
– , and – ,
x x x M x x x N
δ δ δ δ
∈ + ⊂ ∈ + ⊂
Let ( )
1 2
min , .
δ δ δ
= Then
] [ ] [
1 1
– , ,
x x x x M
δ δ δ δ
+ ⊂ − + ⊂
and
] [ ] [
2 2
– , – ,
x x x x N
δ δ δ δ
+ ⊂ + ⊂
⇒ ] [
– ,
x x M N
δ δ
+ ⊂ ∩
⇒ M N
∩ is a nbd of x.
2.1.2 Interior Points of a Set
A point x is an interior point of a set S if S is a nbd of x. In other words, x is an interior point of S if ∃ an
open interval ]a, b[ containing x and contained in S,
i.e., ] , [ .
x a b S
∈ ⊂
Thus a set is a neighbourhood of each of its interior points.
Interior of a Set. The set of all interior points of a set is called the interior of the set. The interior of
a set S is generally denoted by Si
.
Exercise 1. Show that the interior of the set N or I or Q is the null set, but interior of R is R.
Exercise 2. Show that the interior of a set S is a subset of S, i.e., .
i
S S
⊂
2.1.3 Open Set
A set S is said to be open if it is a nbd of each of its points, i.e., for each ,
x S
∈ there exists an open
interval x
I such that
.
x
x I S
∈ ⊂
Thus every point of an open set is an interior point, so that for an open set S, Si
= S.
Evidently, S is open i
S S
⇔ =](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-39-320.jpg)
![Limit Points: Open and Closed Sets 25
Of course the set is not open if it is not a nbd of at least one of its points or that there is at least one
point of the set which is not an interior point.
ILLUSTRATIONS
1. The set R of real numbers is an open set.
2. The set Q of rationals is not an open set.
3. The closed interval [a, b], is not open for it is not a neighbourhood of the end points a and b.
4. The null set φ is open, for there is no point in φ of which it is not a neighbourhood.
5. A non-empty finite set is not open.
6. The set
1
:
∈
n
n
N is not open.
Exercise. Give an example of an open set which is not an interval.
Example 2.5. Show that every open interval is an open set. Or, every open interval is a nbd of each of
its points.
Solution. Let x be any point of the given open interval ]a, b[ so that we have a < x < b.
Let c, d be two numbers such that
a < c < x, and x < d < b
so that we have
] [ ] [
, , .
a c x d b x c d a b
< < < < ⇒ ∈ ⊂
Thus the given interval ]a, b[ contains an open interval containing the point x, and is therefore a nbd
of x.
Hence, the open interval is a nbd of each of its points and is therefore an open set.
Exercise. Show that every point of an open interval is its interior point.
Example 2.6. Show that every open set is a union of open intervals.
Solution. Let S be an open set and xλ a point of S.
Since S is open, therefore ∃ an open interval x
I
λ
for each of its points xλ such that
x
x I S x S
λ
λ λ
∈ ⊂ ∀ ∈
Again the set S can be thought of as the union of singleton sets like { },
xλ
i.e., { },
S xλ
λ∈Λ
= U where Λ is the index set
∴ { } x
S x I S
λ
λ
λ λ
∈Λ ∈Λ
= ⊂ ⊂
U U
⇒ λ
λ∈Λ
= U x
S I
a
c
x
d
b](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-40-320.jpg)
![Limit Points: Open and Closed Sets 27
.
x
x I S F
λ
∈ ⊂ ⊂
Thus the set F contains an open interval containing any point x of F F
⇒ is an open set.
Theorem 2.4. The intersection of any finite number of open sets is open.
Let us consider two open sets S, T.
If ,
S T φ
∩ = it is an open set.
If ,
S T φ
∩ ≠ let x be any point of .
S T
∩
Now x S T x S x T
∈ ∩ ⇒ ∈ Λ ∈ .
⇒ S, T are nbds of x. [ ]
, are open
S T
Q
⇒ S T
∩ is a nbd of x.
But since x is any point of ,
S T
∩ therefore S T
∩ is a nbd of each of its points. Hence, S T
∩ is
open.
The proof may of course be extended to a finite number of sets.
Note. The above theorem does not hold for the intersection of arbitrary family of open sets.
Consider for example the open sets
1 1
– , ,
n
S n
n n
= ∈
N
Their intersection is the set {0} consisting of the single point 0, and this set is not open.
2.2 LIMIT POINTS OF A SET
Definition 1. A real number ξ is a limit point of a set ( )
S ⊂ R if every nbd of ξ contains an infinite
number of members of S.
Thus ξ is a limit point of a set S if for any nbd N of ,
ξ N S
∩ is an infinite set.
A limit point is also called a cluster point, a condensation point or an accumulation point.
A limit point of a set may or may not be a member of the set. Further it is clear from the definition
that a finite set cannot have a limit point. Also it is not neccesary that an infinite set must possess a limit
point. In fact a set may have no limit point, a unique limit point, a finite or an infinite number of limit
points. A sufficient condition for the existence of a limit point is provided by Bolzano-Weierstrass theorem
which is discussed in the next section. The following is another definition of a limit point.
Definition 2. A real number ξ is a limit point of a set ( )
S ⊂ R if every nbd of ξ contains at least one
member of S other than .
ξ
The essential idea here is that the points of S different from ξ get ‘arbitrarily close’ to ξ or ‘pile up’
at .
ξ
Evidently definition 1 implies definition 2. Let us now prove that definition 2 implies definition 1.
x2 x1
ξ δ
− 1 ξ ξ δ
+ 1](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-42-320.jpg)
![28 Principles of Real Analysis
Let ξ be a limit point of the set ( )
S ⊂ R such that every nbd of ξ contains at least one point of
S other than .
ξ Let 1 1
] – , [
ξ δ ξ δ
+ be one such nbd of ξ which contains at least one point, say,
1 of .
x S
ξ
≠
Let 1 2 1
| | .
x ξ δ δ
− = < Now consider the nbd 2 2
] – , [
ξ δ ξ δ
+ of ξ which by def. 2 of a limit
point, must have one point, say, x2 of S other than .
ξ
By repeating the argument with the nbd 3 3
] – , [
ξ δ ξ δ
+ of ξ where 3 2
| – |
x
δ ξ
= and so on, it
follows that the nbd ] – , [ of
i i
ξ δ ξ δ ξ
+ contains an infinity of members of S.
Hence, Def. 2 ⇒ Def. 1.
It is instructive to note that a point ξ is not a limit point of a set S if ∃ even one nbd of ξ not
containing any point of S other than .
ξ
Exercise. Give a bounded set having (i) no limit point, (ii) infinite numbers of limit points.
Derived Sets. The set of all limit points of a set S is called the derived set of S and is denoted by .
'
S
ILLUSTRATIONS
1. The set I has no limit point, for a nbd 1 1
2 2
– ,
m m
+
of ,
m ∈ I contains no point of I other
than m. Thus the derived set of I is the null set .
φ
2. Every point of R is a limit point, for, every nbd of any of its points contains an infinity of
members of R. Therefore .
′ =
R R
3. Every point of the set Q of rationals is a limit point, for, between any two rationals there exist
an infinity of rationals. Further every irrational number is also a limit point of Q for between
any two irrationals there are infinitely many rationals. Thus every real number is a limit point
of Q, so that ′ =
Q R.
4. The set
1
: n
n
∈
N has only one limit point, zero, which is not a member of the set.
5. Every point of the closed interval [a, b] is its limit point, and a point not belonging to the
interval is not a limit point. Thus the derived set [ , ] [ , ].
a b a b
′ =
6. Every point of the open interval ]a, b[ is its limit point. The end points a, b which are not
members of ]a, b[ are also its limit points. Thus
] [ [ ]
, , .
a b a b
′ =
Examples. Obtain the derived sets:
1. { }
: 0 1 ,
x x
≤ <
2. { }
: 0 1, ,
x x x
< < ∈ Q
3. { }
1 1 1
2 2 3
1, – 1, 1 , – 1 , – 1 , ,
K
4.
1
1 : ,
n
n
+ ∈
N](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-43-320.jpg)
![Limit Points: Open and Closed Sets 29
5.
1 1
: , .
m n
m n
+ ∈ ∈
N N
2.2.1 A finite set has no limit point. Also we have seen that an infinite set may or may not have limit
points. We shall now discuss a theorem which sets out sufficient conditions for a set to have limit points.
Bolzano-Weierstrass Theorem (for sets). Every infinite bounded set has a limit point.
Let S be any infinite bounded set and m, M its infimum and supremum respectively. Let P be a set of
real numbers defined as follows:
x P
∈ iff it exceeds at the most a finite number of members of S.
The set P is non-empty, for .
∈
M P Also M is an upper bound of P, for no number greater than or
equal to M can belong to P. Thus the set P is non-empty and is bounded above. Therefore, by the order-
completeness property, P has the supremum, say .
ξ We shall now show that ξ is a limit point of S.
Consider any nbd. ] – , [
ξ ε ξ ε
+ of ,
ξ where 0.
ε >
Since ξ is the supremum of P, ∃ at least one member say η of P such that – .
η ξ ε
> Now η
belongs to P, therefore it exceeds at the most a finite number of members of S, and consequently
– ( )
ξ ε η
< can exceed at the most a finite number of members of S.
Again as ξ is the supremum of P, ξ ε
+ cannot belong to P, and consequently ξ ε
+ must exceed
an infinite number of members of S.
Now –
ξ ε exceeds at the most a finite number of members of S and ξ ε
+ exceeds infinitely many
members of S.
] – , [
ξ ε ξ ε
⇒ + contains an infinite number of members of S.
Consequently ξ is a limit point of S.
2.2.2 Example 2.7. If S and T are subsets of real numbers, then show that
(i) ,
S T S T
′ ′
⊂ ⇒ ⊂ and
(ii) ( )
S T S T
′ ′ ′
∪ = ∪ .
Solution. (i) If S φ
′ = , then evidently .
S T
′ ′
⊂
When ,
S φ
′ ≠ let S
ξ ′
∈ and N be any nbd of .
ξ
⇒ N contains an infinite number of members of S.
But ,
S T
⊂
N
∴ contains infinitely many members of T
⇒ ξ is limit point of T, i.e., .
T
ξ ′
∈
Thus .
S T
ξ ξ
′ ′
∈ ⇒ ∈
Hence .
S T
′ ′
⊂
(ii) Now ( )
S S T S S T ′
′
⊂ ⇒ ⊂
U U
and ( )
T S T T S T ′
′
⊂ ∪ ⇒ ⊂ ∪
Consequently, ( )
S T S T ′
′ ′
∪ ⊂ ∪ ...(1)](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-44-320.jpg)
![30 Principles of Real Analysis
Now we proceed to show that ( ) .
S T S T
′ ′ ′
∪ ⊂ ∪
If ( ) ,
S T φ
′
∪ = then evidently ( ) .
S T S T
′ ′ ′
∪ ⊂ ∪
When ( ) ,
S T φ
′
∪ ≠ let ( ) .
S T
ξ ′
∈ ∪
Now ξ is a limit point of ( ),
S T
∪ therefore, every nbd of ξ contains an infinite number of points
of ( )
S T
∪ ⇒ every nbd of ξ contains infinitely many points of S or T or both.
⇒ ξ is a limit point of S or a limit point of T
⇒ .
S T S T
ξ ξ ξ
′ ′ ′ ′
∈ ∨ ∈ ⇒ ∈ ∪
Thus ( ) .
S T S T
ξ ξ
′ ′ ′
∈ ∪ ⇒ ∈ ∪
Consequently, ( )
S T S T
′ ′ ′
∪ ⊂ ∪ ...(2)
From (1) and (2) it follows that
( )
S T S T
′ ′ ′
∪ = ∪
Thus the derived set of the union = the union of the derived sets.
(ii) Aliter. To show that ( )
S T S T
′ ′ ′
∪ ⊂ ∪
We may show that ( ) .
S T S T
ξ ξ
′ ′ ′
∉ ∪ ⇒ ∉ ∪
Now S T
ξ ′ ′
∉ ∪ implies that ξ does not belong to either.
⇒ ξ is not a limit point of S or of T
∴∃ nbds. 1 2
, of
N N ξ such that N1 contains no point of S other than ξ and N2 contains no point
of T other than possibly .
ξ
Again, since 1 2 1 1 2 2
,
N N N N N N
∩ ⊂ ∩ ⊂ therefore ∃ a nbd. 1 2 of
N N ξ
∩ which contains no
point other than ξ of S or of T and thus of S T
∪ .
⇒ ξ is not a limit point S T
∪ .
⇒ ( )′
∉ ∪
S T
ξ
Thus ( )
S T S T
ξ ξ ′
′ ′
∉ ∪ ⇒ ∉ ∪
so that ( )
S T S T
′ ′ ′
∪ ⊂ ∪
Example 2.8. (i) If S, T are subsets of R, then show that
( )
S T S T
′ ′ ′
∩ ⊂ ∩ .
(ii) Give an example to show that ( )
S T ′
∩ and S T
′ ′
∩ may not be equal.
Solution. (i) Now ( )
S T S S T S
′ ′
∩ ⊂ ⇒ ∩ ⊂
and ( )
S T T S T T
′ ′
∩ ⊂ ⇒ ∩ ⊂
Consequently, ( )
S T S T
′ ′ ′
∩ ⊂ ∩
(ii) Let S = ]1, 2[ and T = ]2, 3[, so that
( ) .
S T S T
φ φ φ
′ ′
∩ = ⇒ ∩ = =](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-45-320.jpg)
![Limit Points: Open and Closed Sets 31
Also [ ] [ ]
1, 2 , 2, 3
S T
′ ′
= =
∴ { }
2 .
S T
′ ′
∩ =
Thus ( ) .
S T S T
′ ′ ′
∩ ≠ ∩
2.3 CLOSED SETS: CLOSURE OF A SET
2.3.1 A real number ξ is said to be an adherent point of a set ( )
S ⊂ R if every nbd of ξ contains
at least one point of S.
Evidently an adherent point may or may not belong to the set and it may or may not be a limit point
of the set.
It follows from the definition that a number S
ξ ∈ is automatically an adherent point of the set, for,
every nbd of a member of the set contains atleast one member of the set, namely the member itself.
Further a number S
ξ ∉ is an adherent point of S only if ξ is a limit point of S, for, every nbd of ξ then
contains atleast one point of S which is other than .
ξ
Thus the set of adherent points of S consists of S and the derived set .
S′
The set of all adherent point of S, called the closure of S, is denoted by ,
S
% and is such that
S S S′
= ∪
% .
ILLUSTRATIONS
1. .
φ
′
= ∪ = ∪ =
I I I I I
%
2. .
′
= ∪ = ∪ =
Q Q Q Q R R
%
3. ′
= ∪ = ∪ =
R R R R R R
%
4. .
φ φ φ φ φ φ
′
= ∪ = ∪ =
%
2.3.2 Closed Sets
A set is said to be closed if each of its limit points is a member of the set.
In other words a set S is closed if no limit point of S exists which is not contained in S. In rough
terms, a set is closed if its points do not get arbitrarily close to any point outside of it.
Thus a set S is closed iff
or .
S S S S
′ ⊂ =
%
Consequently, a closed set is also defined as a set S for which
.
S S
=
%
It should be clearly understood that the concept of closed and open sets are neither mutually exclusive
nor exhaustive. The word not closed should not be considered equivalent to open. Sets exist which are
both open and closed, or which are neither open nor closed. The set consisting of points of ]a, b] is
neither open nor closed.
ILLUSTRATIONS
1. [a, b] is a set which is closed but not open.](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-46-320.jpg)
![32 Principles of Real Analysis
2. The set [0, 1] [2, 3],
∪ which is not an interval, is closed.
3. The null set φ is closed for there exists no limit point of φ which is not contained in .
φ As
was shown earlier, φ is also open.
4. The set R of real numbers is open as well as closed.
5. The set Q is not closed, for .
′ = ⊂
Q R | Q Also it is not open.
6.
1
: n
n
∈
N is not closed, for it has one limit point, 0, which is not a member of the set. Also
it is not open.
7. Every finite set A is a closed set, for its derived set .
A A
φ
′ = ⊂
8. A set A which has no limit point coincides with its closure, for A φ
′ = and .
A A A A
′
= ∪ =
%
2.3.3 Typical Examples
Example 2.9. Show that the set { : 0 1, }
S x x x
= < < ∈ R is open but not closed.
Solution. The set S is the open interval ]0, 1[.
∴ It contains a nbd of each of its points. Hence, it is an open set.
Again every point of S is a limit point. The end points 0 and 1 which are not members of the set are
also limit points. Thus S is not closed.
Example 2.10. Show that the set
1 1 1 1
1, 1, , , , – , ...
2 2 3 3
= − −
S
is neither open nor closed.
Solution. The members of S heap or cluster near zero on both sides of it and every nbd of zero contains
an infinite number of points of S. Thus 0 S
∉ is a limit point ⇒ S is not closed.
Again S is not open for it does not contain any nbd of any of its points. For example, a nbd
1 1 1 1
,
3 100 3 100
− +
of
1
3
is not contained in the set. Hence, the set is not open.
Example 2.11. Show that the set
1 1 1 1
1, 1,1 , 1 ,1 , 1 , ...
2 2 3 3
− − −
is closed but not open.
Solution. 1 and –1 are the only limit points of the set and are in the set. Therefore, the set is closed.
Again all members of the set (except 1, –1) are not the interior points of the set. Thus the set is not
open.
Hence, the set is closed but not open.
The relationship between closed and open sets is brought out by Theorem 2.5 that follows and is
sometimes taken as the definition of a closed set.](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-47-320.jpg)
![34 Principles of Real Analysis
∴ ∃ nbds N1 and N2 of ξ such that
1 2 .
N S N T
φ φ
∩ = ∧ ∩ = ...(1)
Let 1 2 , where .
N N N N
ξ
∩ = ∈
∴ From (1) it follows that
( ) .
N S T N F
φ φ
∩ ∪ = ⇒ ∩ =
Thus ∃ a nbd N of ξ which contains no point of F.
⇒ ξ is not a limit point of F, which is a contradiction.
Hence, no point not belonging to F can be its limit point, and consequently F S T
= ∪ is a closed set.
Remarks 1. The theorem can be extended to the union of a finite number of sets. So we may restate the theorem as:
The union of a finite number of closed sets is closed.
2. We have given an independent proof but the theorem follows from theorem 4 on taking complements.
3. The union of an arbitrary family of closed sets may not always be a closed set. For example, let
1
, 2 , for .
n
S a a n a
n
= + + ∈ Λ ∈
N R
Then ] ]
, 2 ,
n
n
S a a
∈
∪ = +
N
which is not a closed set.
Theorem 2.8. The derived set of a set is closed.
Let S′ be the derived set of a set S.
We have to show that the derived set S′′ of S′ is contained in .
S′
Now if ,
S φ
′′ = i.e., when S′ is either a finite set or an infinite set without limit points, then
S S
φ
′′ ′
= ⊂ and therefore S′ is closed.
When ,
S φ
′′ ≠ let ,
S
ξ ′′
∈ i.e., ξ be a limit point of .
S′
∴ Every nbd N of ξ contains at least one point of .
S
η ξ ′
≠
Again,
S
η η
′
∈ ⇒ is a limit point of S
⇒ every nbd of ,
η N being such a nbd, contains infinitely many points of S.
Thus every nbd N (of )
ξ contains an infinitely many points of S
⇒ ξ is a limit point of S, i.e., .
S
ξ ′
∈
Consequently S S
ξ ξ
′′ ′
∈ ⇒ ∈
∴ ,
S S
⊂
′′ ′ i.e., S′ is a closed set.
Corollary 1. S′′ is closed, and therefore the closure of S′ is ,
S′′ i.e., .
S S′
=
%
Corollary 2. For every set S the closure S
% is closed.
We have simply to show that ( ) .
S S
′ ⊂
% %
Now, ( ) ( ) .
S S S S S S S
′ ′
′ ′ ′′ ′
= ∪ = ∪ = ⊂
% % (ref. § 2.2)](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-49-320.jpg)
![Limit Points: Open and Closed Sets 35
Corollary 3. For every set ,
S S S
=
%
% %
( ) .
S S S S
′
= ∪ =
%
% % % %
Theorem 2.9. The supremum (infimum) of a bounded non-empty set ( ),
⊂
S R when not a member
of S, is a limit point of S.
Let M be the supremum of the bounded set ( ),
S ⊂ R which must exist by the order completeness
property of R. If ,
M S
∉ then for any number 0,
ε > however small, ∃ at least one member x of S such
that
.
M x M
ε
− < <
Thus every nbd of M contains atleast one member x of the set S other than M. Hence M is a limit
point of S.
Corollary. The supremum (infimum) M of a bounded set S is always a member of the closure S
% of S.
When ,
M S
∈
M S M S S S
′
∈ ⇒ ∈ ∪ = %
When ,
M S
∉
M S M S M S S S
′ ′
∉ ⇒ ∈ ⇒ ∈ ∪ = %
Consequently .
M S
∈ %
Theorem 2.10. The derived set of a bounded set is bounded.
Let m, M be the bounds of a set S.
It will now be shown that no limit point of S can be outside the interval [m, M].
Let, if possible, M
ξ > be a limit point of S, and ε be a positive number such that .
M
ε ξ
< −
Then since M is an upper bound of S, no member of S can lie in the interval ] , [,
ξ ε ξ ε
− +
therefore ∃ a nbd of ξ which contains no point of S so that ξ cannot be a limit point of S.
Hence S has no limit point greater than M.
Similarly it can be shown that no limit point of S is less than m.
Hence [ ]
, .
S m M
′ ⊂
Corollary. If S is bounded then so is its closure S
%.
[ ] [ ] [ ]
, , , .
S m M S m M S S S m M
′ ′
⊂ ⇒ ⊂ ⇒ = ∪ ⊂
%
Remark. If supremum M (infimum m) of S is not a member of S, then it is a limit point of S and in view of the above
theorem, it is the greatest (least) member of .
S′
However, if it is a member of S, then it is not necessarily a limit point of S, so that M (or m) may not be a
member of [ , ].
S m M
′ ⊂ Thus M, m may not always be supremum and infimum of S′ but they are always
so for .
S S S′
= ∪
%
For example, for the set
1 1 1 1
1,1, 1 ,1 , 1 , 1 , ...
2 2 3 3
= − − −
S
m M
ξ ε
− ξ ε
+
N](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-50-320.jpg)
![36 Principles of Real Analysis
{ }
1 1
1 , 1 , 1,1
2 2
= − = = −
′
m M S
inf 1 , Sup 1
S m S M
′ ′
= − ≠ = ≠
but
inf , Sup .
S m S M
= =
% %
Theorem 2.11. The derived set S′ of a bounded infinite set ( )
S ⊂ R has the smallest and the
greatest members.
Since the set S is bounded, therefore S′ is also bounded. Also S′ is non-empty, for by Bolzano-
Weierstrass theorem S has at least one limit point.
Now S′ may be finite or inifinite.
When ( )
S φ
′ ≠ is finite, evidently it has the greatest and the least members.
When S′ is infinite, being bounded set of real numbers, by order-completeness property of R, it has
the supremum G and the infimum g.
It will now be shown that G, g are limit points of S,
i.e., ,
G S g S
′ ′
∈ ∈
Let us first consider G.
] , [, > 0,
G G
ε ε ε
− + be any nbd of G.
Now G being the supremum of ,
S′ ∃ at least one member ξ of S′ such that .
G G
ε ξ
− < ≤
Thus ] , [
G G
ε ε
− + is a nbd of .
ξ
But ξ is a limit point of S, so that ] , [
G G
ε ε
− + contains an infinity of members of S
⇒ any ] , [
nbd G G
ε ε
− + of G contains an infinite number of members of S
⇒ G is a limit point of S G S′
⇒ ∈
Similarly, it can be shown that g S
∈ ′.
Thus G S g S
∈ ′ ∈ ′
and , being supremum and infimum of ′
S , are the greatest and the smallest
members of ′
S .
The theorem may be restated as:
Every bounded infinite set has the smallest and the greatest limit points.
The smallest and greatest limit points of a set are called the lower and upper limits of
indetermination or simply the lower and upper limits respectively of the set.](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-51-320.jpg)
![Real Sequences 39
S K
n ≤ ∀ ∈
n N
Bounded below sequences
A sequence {Sn} is said to be bounded below if there exists a real number k such that
S k n
n ≥ ∀ ∈N.
Bounded sequences
A sequence is said to be bounded when it is bounded both above and below. K and k are respectively
the upper and the lower bounds of the sequence.
Evidently a sequence is bounded iff its range is bounded. Also the bounds of the range are the
bounds of the sequence.
3.2.3 Convergence of Sequences
Definition 1. A sequence {Sn} is said to converge to a real number l (or to have the real number l as
its limit) if for each A > 0, there exists a positive integer m (depending on ε ) such that ,
n
S l ε
− < for
all n m
≥ .
The fact is expressed by saying that the terms approach the value l or tend to l as n becomes larger
and larger. The same thing expressed in symbols is
S l n
n ® ® ¥
for
or lim .
n
n
S l
®¥
=
The definition ensures that
(i) From some stage onwards the difference between Sn and l can be made less than any preassigned
positive number ε, however small, i.e., given any positive real number ε, no matter however
small, ∃ a positive integer m (finite) such that mth term onwards, Sn becomes and remains
arbitrarily close to l, i.e. l is a limit point of the sequence.
(ii) For any ε > 0, at the most a finite number of terms (depending on the choice of ε ) of the
sequence can lie outside ] , [,
l l
− +
ε ε i.e. there is at the most a finite number of n’s for which
S l S l
n n
≤ − ≥ +
ε ε
and .
(iii) Since l S l
n
− < < +
ε ε for all n m
≥ , therefore S l
n < + ε , for infinite number of terms,
i.e., infinite number of terms lie to the left of l + ε, or to the right of l − ε.
It may therefore be observed that if we can find even one ε > 0 for which infinitely many
terms of the sequence lie outside ] , [,
l l
− +
ε ε then the sequence cannot converge to l.
3.2.4 Some Theorems
Theorem 3.1. Every convergent sequence is bounded.
Let a sequence {Sn} converge to the limit l.
Let ε > 0 be a given number, so that ∃ a positive integer m such that
S l n m
n − < ∀ ≥
ε
⇔ − < < + ∀ ≥
l S l n m
n
ε ε .
Let g l S S Sm
= − −
min , , , , .
ε 1 2 1
...
n s](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-54-320.jpg)
![regarded as the chief means for attaining the spiritual end of the
monastic life.” He calls his Rule “a very little rule for beginners”—
minima inchoationis regula, and says that though there may be in it
some things “a little severe,” still he hopes that he will establish
“nothing harsh, nothing heavy.” The most cursory comparison
between this new Rule and those which previously existed will make
it abundantly clear that St. Benedict’s legislation was conceived in a
spirit of moderation in regard to every detail of the monastic life.
Common-sense, and the wise consideration of the superior in
tempering any possible severity, according to the needs of times,
places, and circumstances were, by his desire, to preside over the
spiritual growth of those trained in his “school of divine service.”
In addition to this St. Benedict broke with the past in another and
not less important way, and in one which, if rightly considered and
acted upon, more than compensated for the mitigation of corporal
austerities introduced into his rule of life. The strong note of
individualism characteristic of Egyptian monachism, which gave rise
to what Dom Butler calls the “rivalry in ascetical achievement,” gave
place in St. Benedict’s code to the common practices of the
community, and to the entire submission of the individual will, even
in matters of personal austerity and mortification, to the judgment of
the superior.
“This two-fold break with the past, in the elimination of
austerity and in the sinking of the individual in the
community, made St. Benedict’s Rule less a development
than a revolution in monachism. It may be almost called a
new creation; and it was destined to prove, as the
subsequent history shows, peculiarly adapted to the new
races that were peopling Western Europe.”[7]
We are now in a position to turn to England. When, less than half a
century after St. Benedict’s death, St. Augustine and his fellow
monks in a.d. 597 first brought this Rule of Life to our country, a
system of monasticism had been long established in the land. It was
Celtic in its immediate origin; but whether it had been imported](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-56-320.jpg)
![early Irish saints to show that the holder of the office was certainly
treated with special dignity and honour.
The Celtic monastic system was apparently in vogue among the
remnant of the ancient British Church in Wales and the West Country
on the coming of St. Augustine. Little is known with certainty, but as
the British Church was Celtic in origin it may be presumed that the
Celtic type of monachism prevailed amongst the Christians in this
country after the Saxon conquest. Whether it followed the distinctive
practice of Irish monasticism in regard to the position of the abbot
and the subject bishops may perhaps be doubted, as this does not
appear to have been the practice of the Celtic Church of Gaul, with
which there was a close early connection.
It has usually been supposed that the Rule of St. Columbanus
represented the normal life of a Celtic monastery, but it has been
lately shown that, so far as regards the Irish or Welsh houses, this
Rule was never taken as a guide. It had its origin apparently in the
fact that the Celtic monks on the Continent were induced, almost in
spite of themselves, to adopt a mitigated rule of life by their close
contact with Latin monasticism, which was then organising itself on
the lines of the Rule of St. Benedict.[8] The Columban Rule was a
code of great rigour, and “would, if carried out in its entirety, have
made the Celtic monks almost, if not quite, the most austere of
men.” Even if it was not actually in use, the Rule of St. Columbanus
may safely be taken to indicate the tendencies of Celtic monasticism
generally, and the impracticable nature of much of the legislation
and the hard spirit which characterises it goes far to explain how it
came to pass that whenever it was brought face to face with the
wider, milder, and more flexible code of St. Benedict, invariably,
sooner or later, it gave place to it. In some monasteries, for a time,
the two Rules seem to have been combined, or at least to have
existed side by side, as at Luxeuil and Bobbio, in Italy, in the seventh
century; but when the abbot of the former monastery was called
upon to defend the Celtic rule, at the Synod of Macon in a.d. 625,](https://image.slidesharecdn.com/16777387-250606180729-f8601a23/85/Principles-Of-Real-Analysis-2nd-Edition-S-C-Malik-58-320.jpg)
Principles Of Real Analysis 2nd Edition S C Malik
- 1. Principles Of Real Analysis 2nd Edition S C Malik download https://ebookbell.com/product/principles-of-real-analysis-2nd- edition-s-c-malik-33554774 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Principles Of Real Analysis Measure Integration Functional Analysis And Applications Hugo D Junghenn https://ebookbell.com/product/principles-of-real-analysis-measure- integration-functional-analysis-and-applications-hugo-d- junghenn-7161044 Direct Analysis In Real Time Mass Spectrometry Principles And Practices Of Dartms 1st Edition Yiyang Dong https://ebookbell.com/product/direct-analysis-in-real-time-mass- spectrometry-principles-and-practices-of-dartms-1st-edition-yiyang- dong-7034638 Principles Of Real Estate Management Nicholas Dunlap https://ebookbell.com/product/principles-of-real-estate-management- nicholas-dunlap-48507368 Principles Of Real Estate Syndication Samuel K Freshman https://ebookbell.com/product/principles-of-real-estate-syndication- samuel-k-freshman-48777098
- 3. Principles Of Real Estate Management 16th Edition Tulie Oconnor Ed https://ebookbell.com/product/principles-of-real-estate- management-16th-edition-tulie-oconnor-ed-48957458 Principles Of Real Estate Management Nicholas Dunlap https://ebookbell.com/product/principles-of-real-estate-management- nicholas-dunlap-231850218 Pennsylvania Principles Of Real Estate Practice Paperback Stephen Mettling David Cusic https://ebookbell.com/product/pennsylvania-principles-of-real-estate- practice-paperback-stephen-mettling-david-cusic-7437714 Relentless Pursuit The Foundations And Principles Of Real Estate Investing Nalley https://ebookbell.com/product/relentless-pursuit-the-foundations-and- principles-of-real-estate-investing-nalley-75758184 Principles Of California Real Estate Kathryn Hauptdavid Rockwell Kathryn Hauptdavid Rockwell https://ebookbell.com/product/principles-of-california-real-estate- kathryn-hauptdavid-rockwell-kathryn-hauptdavid-rockwell-33188580
- 8. S C MALIK Former Senior Lecturer Department of Mathematics SGTB Khalsa College, University of Delhi Delhi, India PRINCIPLES OF REAL ANALYSIS NEW ACADEMIC SCIENCE The Control Centre, 11 A Little Mount Sion Tunbridge Wells, Kent TN1 1YS, UK www.newacademicscience.co.uk e-mail: info@newacademicscience.co.uk New Academic Science Limited (SECOND EDITION)
- 9. Copyright © 2013 by New Academic Science Limited The Control Centre, 11 A Little Mount Sion, Tunbridge Wells, Kent TN1 1YS, UK www.newacademicscience.co.uk • e-mail: info@newacademicscience.co.uk ISBN : 978 1 781830 49 9 All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical,withoutthewrittenpermissionof thecopyrightowner. British Library Cataloguing in Publication Data A Catalogue record for this book is available from the British Library Every effort has been made to make the book error free. However, the author and publisher have no warrantyofanykind,expressed orimplied,withregardtothedocumentationcontainedinthisbook.
- 10. Preface The aim has been to provide a development of the subject-matter which is honest, rigorous and at the same time not too pedantic. Most of the hard theorems which are either omitted or treated rather skimpily in many texts in advanced analysis are proved with care. Some of them are ordinarily considered too difficult for a standard course in calculus but too elementary for a course of analysis. With the inclusion of these theorems the book attempts to fill the gap and to make the transfer from elementary calculus to advanced course in analysis as smooth as possible. The book starts with a quick review of the essential properties of rational numbers. Using Dedekind’s Cut the properties of real numbers are established. This foundation supports the subsequent chapters: topological framework, sequences and series of numbers, continuity, differentiation, elementary functions, Riemann integration, with a quick look at the Riemann-Stieltjes integral and finally the functions of several variables and the implicit functions. The material on functions of two variables is given with more details and with more examples and motivation than is usually done. Cantor’s theory of the construction of real numbers appears in an appendix to the book which may be studied and enjoyed whenever time seems ripe. Because of the narrow compass of the book, many important topics had to be omitted which could perhaps be presented in a second volume, Mathematical Analysis for advanced studies. A large number of examples, taken mostly from the question papers of different universities and from his own class notes, have been graded and supplied with answers. The majority of them are straight forward; hints for harder ones are occasionally given. Examples to illustrate the general theory or to show where it breaks down follow every important principle. This and the remarks and notes added here and there should help in fixing the ideas better. Effort has also been made to give the proofs of a numbers of theorems in a somewhat modified form. During the study of the subject and the preparation of lecture notes for more than three decades of my university teaching (where from the present book has taken shape) I have made use of the standard works of a large number of authors including Bromwich, Carslaw, Courant, Ferrar, Gibson, Hyslop, Knopp, Landau and Phillips. I am fully aware that I have benefited largely from them but I cannot resolve, how much is due to each. I owe much to them and I gladly take this opportunity to acknowledging my indebtedness to them. Finally I may add that the author as also the publishers will welcome all suggestions for improvement of the book. S.C. MALIK
- 12. Contents vii Contents Preface v 1 REAL NUMBERS 1–22 1.1 Introduction 1 1.2 Field Structure and Order Structure 7 1.3 Bounded and Unbounded Sets: Supremum, Infimum 10 1.4 Completeness in the Set of Real Numbers 13 1.5 Absolute Value of a Real Number 18 2 LIMIT POINTS: OPEN AND CLOSED SETS 23–36 2.1 Introduction 23 2.2 Limit Points of a Set 27 2.3 Closed Sets: Closure of a Set 31 3 REAL SEQUENCES 37–69 3.1 Functions 37 3.2 Sequences 37 3.3 Limit Points of a Sequence 40 3.4 Convergent Sequences 43 3.5 Non-Convergent Sequences (Definitions) 44 3.6 Cauchy’s General Principle of Convergence 48 3.7 Algebra of Sequences 51 3.8 Some Important Theorems 55 3.9 Monotonic Sequences 61 4 INFINITE SERIES 70–104 4.1 Introduction 70 4.2 Positive Term Series 74 4.3 Comparison Tests for Positive Term Series 76 4.4 Cauchy’s Root Test 82 4.5 D’Alembert’s Ratio Test 83 4.6 Raabe’s Test 85 4.7 Logarithmic Test 89
- 13. viii Contents 4.8 Integral Test 91 4.9 Gauss’s Test 93 4.10 Series with Arbitrary Terms 98 5 FUNCTIONS WITH INTERVAL AS DOMAIN (I) 105–129 5.1 Limits 105 5.2 Continuous Functions 114 5.3 Functions Continuous on Closed Intervals 122 5.4 Uniform Continuity 126 6 FUNCTIONS WITH INTERVAL AS DOMAIN (II) 130–160 6.1 The Derivative 130 6.2 Continuous Functions 132 6.3 Increasing and Decreasing Functions 136 6.4 Darboux’s Theorem 139 6.5 Rolle’s Theorem 140 6.6 Lagrange’s Mean Value Theorem 141 6.7 Cauchy’s Mean Value Theorem 143 6.8 Higher Order Derivatives 150 7 APPLICATIONS OF TAYLOR’S THEOREM 161–180 7.1 Extreme Values (Definitions) 161 7.2 Indeterminate Forms 167 8 ELEMENTARY FUNCTIONS 181–193 8.1 Introduction 181 8.2 Power Series 181 8.3 Exponential Functions 183 8.4 Logarithmic Functions 185 8.5 Trigonometric Functions 188 9 THE RIEMANN INTEGRAL 194–236 9.1 Introduction 194 9.2 Definitions and Existence of the Integral 194 9.3 Refinement of Partitions 201 9.4 Darboux’s Theorem 202 9.5 Conditions of Integrability 204 9.6 Integrability of the Sum and Difference of Integrable Functions 207 9.7 The Integral as a Limit of Sums (Riemann Sums) 215
- 14. Contents ix 9.8 Some Integrable Functions 222 9.9 Integration and Differentiation (The Primitive) 226 9.10 The Fundamental Theorem of Calculus 227 9.11 Mean Value Theorems of Integral Calculus 231 9.12 Integration by Parts 233 9.13 Change of Variable in an Integral 234 9.14 Second Mean Value Theorem 235 10 THE RIEMANN-STIELTJES INTEGRAL 237–254 10.1 Definitions and Existence of the Integral 237 10.2 A Condition of Integrability 240 10.3 Some Theorems 241 10.4 A Definition (Integral as a Limit of Sum) 245 10.5 Some Important Theorems 251 11 FUNCTIONS OF SEVERAL VARIABLES 255–321 11.1 Explicit and Implicit Functions 255 11.2 Continuity 264 11.3 Partial Derivatives 268 11.4 Differentiability 271 11.5 Partial Derivatives of Higher Order 279 11.6 Differentials of Higher Order 286 11.7 Functions of Functions 288 11.8 Change of Variables 295 11.9 Taylor’s Theorem 306 11.10 Extreme Values: Maxima and Minima 310 11.11 Functions of Several Variables 315 12 IMPLICIT FUNCTIONS 322–345 12.1 Definition 322 12.2 Jacobians 326 12.3 Stationary Values under Subsidiary Conditions 333 Appendix I—Theorems on Rearrangement of Terms and 346–354 Tests for Arbitrary Series 1. Tests for Arbitrary Term Series 346 2. Rearrangement of Terms 349
- 15. x Contents Appendix II—Cantor’s Theory of Real Numbers 355–368 1. Introduction 355 2. Sequences of Rational Numbers 355 3. Real Numbers 357 4. Addition and Multiplication in R 358 5. Order in R 361 6. Real Rational and Irrational Numbers 364 7. Some Properties of Real Numbers 365 8. Completeness in R 366 Bibliography 369–371 Index 373–375
- 16. CHAPTER 1 Real Numbers 1.1 INTRODUCTION In school algebra and arithmetic we usually deal with two fundamental operations viz. addition and multiplication and their inverse operations, subtraction and division respectively. These operations are related to a certain class of ‘numbers’ which will be described more precisely in the following sections. The basic difference between ‘elementary mathematics’ and ‘higher mathematics’, which begins at the college level, consists in introducing the all important notion of limit which is very intimately related to the intuitive idea of nearness or closeness and which cannot be described in terms of the operations of addition and multiplication. The notion of limit comes into play in situations where one quantity depends on another varying quantity and we have to know the behaviour of the first when the second is arbitrarily close to a fixed given value. In order to illustrate our point in relation to a practical situation, consider the question of determining the velocity of the planet earth at a particular instant during its motion round the sun assuming that the path of its motion round the sun and its position on this path at any instant are known. We cannot determine the velocity of earth without taking recourse to the notion of limit and indeed we need the notion of limit even in defining the concept of ‘velocity’ of a moving object which is not moving with a uniform speed. The purpose of this illustration is simply to indicate that there are numerous situations where the methods of elementary algebra prove quite inadequate for the purpose of solving or even formulating a problem, and we are forced to evolve new concepts and methods. The notion of limit is one such concept. For the proper understanding of the notion of limit and its importance, it is absolutely desirable that the reader be familiar with the true nature and important properties of ‘real numbers’. Starting with natural numbers, we shall briefly and intuitively introduce in the following sections the concept of rational numbers and irrational numbers which together form the system of real numbers describing in the process the important properties possessed by these numbers. The branch of mathematics called real analysis deals with problems which are closely connected with the notion of ‘limit’ and some other notions, such as the operations of ‘differentiation’ and ‘integration’ which are directly dependent on the concept of limit when all these operations are confined to the domain of real numbers. It is difficult to say anything more precise at this stage. The interested reader will certainly have a clearer and precise understanding of this important branch of mathematics as he systematically studies this work. 1.1.1 Real Numbers The system of real numbers has evolved as a result of a process of successive extensions of the system of natural numbers (i.e., the positive whole numbers). It may be remarked here that the extension became absolutely inevitable as the science of Mathematics developed in the process of solving problems from
- 17. 2 Principles of Real Analysis other fields. Natural numbers came into existence when man first learnt counting which can also be viewed as adding successively the number 1 to unity. If we add two natural numbers, we get a natural number—but the inverse operation of subtraction is not always possible if we limit ourselves to the domain of natural numbers only. For instance, there is no natural number which added to 8 will give us 3. In other words, 8 cannot be subtracted from 3 within the system of natural numbers. In order that the operation of subtraction (i.e., operation inverse of addition) be also performed without any restriction, it became necessary to enlarge the system of natural numbers by introducing the negative integers and the number zero. Thus to every natural number n corresponds a unique negative integer designated –n and called the additive inverse of n, and there is a number zero, written 0, such that n + (–n) = 0, and n + 0 = n for every natural number n. Also n is the inverse of –n. The negative integers, the number 0, and the natural numbers (i.e., the positive integers) together constitute what is known as the system of integers. Similarly, to make division always possible, zero being an exception, the concept of fractions, positive and negative, was introduced. Division by zero, however, cannot be defined in a meaningful and consistent manner. The system so extended, including integers and fractions both positive and negative, and the number zero, is called the system of rational numbers. Thus, every rational number can be represented in the form p/q where p and q are integers and 0. q ≠ We know that the result of performing any one of the four operations of arithmetic (division by zero being, of course, excluded) in respect of any two rational numbers, is again a rational number. So long as mathematics was concerned with these four operations only, the system of rational numbers was sufficient for all purposes but the process of extracting roots of numbers (e.g., square-root of 2, cube-root of 7, etc.), as also the desirability of giving a meaning to non-terminating and non-recurring decimals, necessitated a further extension of the number system. There were lengths which could not be measured in terms of rational numbers, for instance the length of the diagonal of a square whose sides are of unit length, cannot be measured in terms of rational numbers. In fact, this is equivalent to saying that there is no rational number whose square is equal to 2. In order to be able to answer such questions, the system of rational numbers had to be further enlarged by introducing the so called irrational numbers. It is beyond the scope of this book to discuss systematically the definition of irrational numbers in terms of rational numbers. Numbers like 3 2, 7, π (i.e., the ratio of the circumference to the diameter of a circle) with which the reader is already familiar are examples of irrational numbers. Rational numbers and irrational numbers together constitute what is known as the system of real numbers. Though the real number system cannot be extended in a way in which a rational number system is extended but it can be used to develop another system, called the system of complex numbers. But since real analysis is not concerned with complex numbers, we shall have nothing to do with complex numbers in this book. For the sake of brevity and clarity of exposition, and because the notion of set is fundamental to all branches of mathematics, we start with the algebra of sets. 1.1.2 Sets A set is a well defined collection of objects. In other words, an aggregate or class of objects having a specified property in common which enables us to tell whether any given object belongs to it or not. The individual objects of the set are called members or elements of the set. Capital letters A, B, C, etc., are generally used to denote the sets while small letters a, b, c, etc., for elements. If x is a member of a set A, then we write x A ∈
- 18. Real Numbers 3 and read it as ‘x belongs to A’ or ‘x is an element of A’ or ‘x is a member of A’ or simply ‘x is in A’. If x is not a member of A, then we write x A ∉ and read it as ‘x does not belong to A’. Some Typical Sets N: the set of natural numbers, I: the set of integers, I+ : the set of positive integers, I– : the set of negative integers, Q: the set of rational numbers, R: the set of real numbers. There are two methods which are in common use to denote a set. (i) A set may be described by listing all its elements. (a) Set S has elements a, b, c, then we write S = {a, b, c} (b) Set V of vowels in the English alphabets V = {a, e, i, o, u} (ii) A set may be described by means of a property which is common to all its elements. (a) The set S of all elements x which have the property P(x) S = {x: P(x)} (b) The set B of natural numbers { : } B n n = ∈ N A Null Set. A set having no element. Sometimes the defining property of a set is such that no object can satisfy it, so that the set remains empty. Such a set is called a null set, an empty set or a void set, and is generally denoted by the Danish letter φ or { }. Thus φ = {n : n is a natural number less than 1} { : } x x x φ = ≠ 1.1.3 Equality of Sets Two sets are said to be equal when they consist of exactly the same elements. Thus, sets P, Q are equal (P = Q) if every element of P is an element of Q and every element of Q is also an element of P. Thus, {a, b, c} = {b, a, c} {4, 5, 6, 7, 8, 9} { : 3 10, } n n n N = < < ∈ It is to be noted that in writing a set, an element occurs only once but the order in which the elements of a set are written is immaterial. A set is finite or infinite according to as the number of element in it is finite or infinite.
- 19. 4 Principles of Real Analysis 1.1.4 Notation , , , , , , ~ ∀ ∃ ⇒ ⇔ ∧ ∨ These symbols borrowed from mathematical logic help in a neat and brief exposition of the subject and so we shall describe them here briefly. (i) ∀ stands for ‘for all’ or ‘for every’. The statement x < y, x S ∀ ∈ means, x is less than y for all members of S, i.e., all members of S are less than y. (ii) ∃ stands for ‘there exists’. (iii) ⇒ stands for ‘implies that’. If P and Q are two statements then P Q ⇒ means that the statement P implies the statement Q i.e., if P is true, then Q is also true. Thus 2 5 25 x x = ⇒ = || and || || AB CD CD EF AB EF ⇒ If the statements P and Q are such that P implies Q and Q implies P, then we write P Q ⇔ (both ways implication) Thus for real numbers x, y 0 0 or 0 xy x y = ⇔ = = (iv) ∧ stands for ‘and’ ∨ stands for ‘or’ The statement P Q ∧ holds when both the statements P and Q hold, but the statement R S ∨ can hold when either R holds or S holds, i.e., R S ∨ holds when at least one of R and S holds. Thus ( ) ( ) 3 5 0 3 5 x x x x − − < ⇒ > ∧ < 2 1 1 1 x x x = ⇒ = ∨ = − (v) Negation ~ stands for ‘not’. If P is a statement then ~ P is negation of P. In other words, ~ P denotes ‘not P’. Thus when P holds, ~ P cannot hold and vice versa. ~ P P ∧ is always false, but ~ P P ∨ is always true. 1.1.5 Subsets If A and B are two sets such that each member of A is also a member of B i.e., , x A x B ∈ ⇒ ∈ then A is called a subset of B (or is contained in B) and we write . A B ⊂
- 20. Real Numbers 5 This is sometimes expressed by saying that B is a Superset of A (or contains A) and we write . B A ⊃ Thus, if A is a subset of B then there is no element in A which is not in B, i.e., . y B y A ∉ ⇒ ∉ Consequently, the null set φ is a subset of every set, and A A ⊂ for every set A. Thus, if A B ⊂ and , B A ⊂ we write A = B. A B ⊆ allows for the possibility that A and B might be equal. If A is a subset of B and is not equal to B, we say that A is a proper subset of B (or is properly contained in B). Thus A is a proper subset of B if every member of A is a member of B and there exists at least one member of B which is not a member of A. Two sets A and B are said to be comparable if either or , A B A B ⊃ ⊂ otherwise they are not comparable. 1.1.6 Union and Intersection of Sets Union. If A and B are two sets, then the set consisting of all those elements which belong to A or to B or to both, is called the union of A and B and is denoted by . A B 7 Clearly , A A φ = 7 A A A = 7 and . A B B A = 7 7 Intersection. If A and B are two sets then, the set consisting of all those elements which belong to both A and B, is called the intersection of A and B and is denoted by . A B 1 Clearly , A φ φ = 1 A A A = 1 . A B B A = 1 1 Thus, A B 1 consists of elements which are common to A and B. Two sets A and B are said to be Disjoint if they have no common element, i.e., . A B φ = 1 1.1.7 Union and Intersection of an Arbitrary Family The operations of forming unions and intersections are primarily binary operations, that is, each is a process which applies to a pair of sets and yields a third. We emphasize this by the use of parentheses to indicate the order in which the operations are to be performed, as in 1 2 3 ( ) , A A A 7 7 where the parentheses direct us first to unite A1 and A2, and then to unite the result with A3. Associativity makes it possible to dispense with parentheses in an expression like this and to write 1 2 3 , A A A 7 7 where we understand that these sets are to be united in any order and that the order in which the operations are performed is irrelevant. Similar remarks apply to 1 2 3. A A A 1 1 Furthermore, if {A1, A2, ..., An} is any finite class of sets, then we can form 1 2 1 2 ... and ... n n A A A A A A 7 7 7 1 1 1
- 21. 6 Principles of Real Analysis in much the same way without any ambiguity of meaning whatever. In order to shorten the notation, we let I = {1, 2, ..., n} be the set of subscripts which index the set under consideration. I is called the Index Set. We then compress the symbols and write 1 1 and or and n n i i i i i I i I i i A A A A ∈ ∈ = = 7 1 7 1 It is often necessary to form unions and intersections of large (really large) class of sets. Let Λ be a set and { : } Aλ λ∈ Λ an entirely arbitrary class or familyF of sets which contains a set Aλ for each λ in . Λ Then { : for at least one in } A x x A λ λ λ λ ∈ Λ = ∈ Λ 7 { : for every in } A x x A λ λ λ λ ∈ Λ = ∈ Λ 1 define the union and intersection of an arbitrary familyF . Λ is called the Index set. If the class {Ai } consists of a sequence of sets, {Ai } = {A1 , A2 , ...} then their union and intersection are often written in the form 1 1 and . i i i i A A ∞ ∞ = = 7 1 1.1.8 Universal Set In any discussion of sets, all sets are usually assumed to be subsets of a set, called the universal set (usually denoted by 7 ). In our present discussion, however, the set R of real numbers can serve as the universal set. 1.1.9 Difference Set, Complement of a Set If A and B are two sets then the set consisting of those elements of B which do not belong to A is called the difference set of A and B and is denoted by B ~ A or B – A. If, however, A is a subset of B then B – A is called the complement of A in B or complement of A with respect to B. Complement of A in the universal set 7 is called the complement of A and is denoted by A or ~A. 1.1.10 Compositions We shall be dealing mainly with number sets and so we define only two types of compositions in the sets. An Addition Composition is defined in a set S if to each pair of members a, b of S there corresponds a member a + b of S. Similarly, a Multiplication Composition is defined in S if to each pair of members a, b of S there corresponds a member ab of S. A set is said to possess an algebraic structure if the two compositions of Addition and Multiplication are defined in the set. Subtraction and Division may be defined as inverse operations of addition and multiplication respectively. Let , . a b S ∈
- 22. Real Numbers 7 Subtraction: a – b may be expressed as a + (–b) when . b S − ∈ Division: The quotient / ( 0) a b b ≠ may be put as 1/ a b ⋅ or ab–1 when 1/b or 1 . b S − ∈ 1.2 FIELD STRUCTURE AND ORDER STRUCTURE 1.2.1 Field Structure A set S is said to be a field if two compositions of Addition and Multiplication defined in it be such that , , a b c S ∀ ∈ the following properties are satisfied: A-1. Set S is closed for addition, , a b S a b S ∈ ⇒ + ∈ A-2. Addition is commutative, a + b = b + a A-3. Addition is associative, (a + b) + c = a + (b + c) A-4. Additive identity exists, i.e., ∃ a member 0 in S such that a + 0 = a A-5. Additive inverse exists, i.e., to each element a S ∈ there exists an element a S − ∈ such that a + (– a) = 0 M-1. S is closed for multiplication, , a b S ab S ∈ ⇒ ∈ M-2. Multiplication is commutative, ab = ba M-3. Multiplication is associative, (ab)c = a(bc) M-4. Multiplicative identity exists i.e.,∃ a member 1 in S such that 1 a a ⋅ = M-5. Multiplicative inverse exists i.e., to each 0 , a S ≠ ∈ ∃ an element 1 a S − ∈ such that aa–1 = 1 AM. Multiplication is distributive with respect to addition, i.e., a(b + c) = ab + ac Thus, a set S has a field structure if it possesses the two compositions of addition and multiplication and satisfies the eleven properties listed above. 1.2.2 Order Structure Ordinarily the order relation does not exist between the members of a general field, but as we are to deal with the field of real numbers, we can speak of one number being ‘greater than’ (or less than) the other.
- 23. 8 Principles of Real Analysis A field S is an ordered field if it satisfies the following properties: O-1. Law of Trichotomy: For any two elements , , a b S ∈ one and only one of the following is true. a > b, a = b, b > a O-2. Transitivity: , , , a b c S ∀ ∈ a b b c a c > ∧ > ⇒ > O-3. Compatibility of Order Relation with Addition Composition: , , , a b c S ∀ ∈ a b a c b c > ⇒ + > + O-4. Compatibility of Order Relation with Multiplication Composition: , , , a b c S ∀ ∈ 0 a b c ac bc > ∧ > ⇒ > 1.2.3 It may be seen that the set Q of rational numbers and the set R of real numbers are ordered fields while the set N of natural numbers and the set I of integers are not fields. (i) The Set N of Natural Numbers 1, 2, 3, 4, ... The sum or product of any two members is easily seen to be a member of N, so that the set possesses two compositions of addition and multiplication, i.e., the set N possesses an algebraic structure. However, it does not satisfy all the properties of a field (it does not possess additive identity, additive inverse and multiplicative inverse) and hence the set of natural numbers is not a field. However, it has an order structure compatible with the algebraic structure. (ii) The Set I of Integers ..., –3, –2, –1, 0, 1, 2, 3, ... It may be easily seen that the set possesses an algebraic structure but does not satisfy all the properties of a field. (M-5 Multiplicative inverses do not exist.) Hence, the set of integers is not a field. However, it has an order structure compatible with the algebraic structure. (iii) The Set Q of Rational Numbers A rational number is of the form p/q, where p and q are integers and 0. q ≠ Evidently, the set Q of rational numbers includes the set of integers. A real number which is not rational (i.e., cannot be expressed as p/q) is called an irrational number. The set R of real numbers consists of rational and irrational numbers. The sets Q and R satisfy all the properties (§ 2.1) of a field and are therefore fields. In addition to this, both these fields satisfy the four properties 0–1 to 0–4 (§ 2.2) of order and hence form ordered fields. 1.2.4 Up to this stage we have discussed two properties—the field property and the order structure property. We have found that both the sets, the set R of real numbers and the set Q of rational numbers possess these properties. However, there is a property called the property of completeness which is possessed only by the set of real numbers and this distinguishes it from other sets of numbers. Let us, now, consider some notions and examples which will facilitate the study of that property.
- 24. Real Numbers 9 1.2.5 Example 1.1. Show that there is no rational number whose square is 2. Solution. Let, if possible, there exist a rational number p/q, where 0 q ≠ and p, q are integers prime to each other (i.e., having no common factor) whose square is equal to 2, i.e., (p/q)2 = 2 or p2 = 2q2 ...(1) Now q is an integer and so is 2q2 . Thus, p2 is an integer divisible by 2. As such p must be divisible by 2, for otherwise p2 would not be divisible by 2. Let p = 2m, where m is an integer. Then, from (1), 2m2 = q2 ...(2) Thus, it follows that q is also divisible by 2. Hence, p and q are both divisible by 2 which contradicts the hypothesis that p and q have no common factor. Thus, there exists no rational number whose square is 2. Example 1.2. Show that 8 is not a rational number. Solution. Let, if possible, 8 be the rational number p/q, where 0 q ≠ and p, q are positive integers prime to each other, so that 8 / p q = But 2 8 3 < < ∴ 2 / 3 2 3 p q q p q < < ⇒ < < or 0 2 p q q < − < Thus, p – 2q is a positive integer less than q, so that 8 ( 2 ) p q − or p/q (p – 2q) is not an integer. But ( ) ( ) 2 8 2 / 2 2 p p q p q p q p q − = − = − 2 2 2 8 2 , p q p q p q = − = − which is an integer. ⇒ 8 ( 2 ) p q − is an integer. Thus we arrive at a contradiction. Hence, 8 is not a rational number. Remark. We have considered n (n not a perfect square), first when n was a prime and then n as a composite number. The procedures shown are typical and may be adopted under similar situations. Exercise. Show that there is no rational number whose square is (i) 3 (ii) 5 (iii) 6. 1.2.6 Intervals—Open and Closed We shall now define different types of intervals. Open Interval. If a and b be two real numbers such that a < b then the set {x : a < x < b}
- 25. 10 Principles of Real Analysis consisting of all real numbers between a and b (excluding a and b) is called an open interval and is denoted by ]a, b[. Closed Interval. The set { : } ≤ ≤ x a x b consisting of a, b and all real numbers lying between a and b is called a closed interval and is denoted by [a, b]. Semi-closed or Semi-open Intervals The intervals ] , ] { : } = < ≤ a b x a x b and, [ , [ { : } = ≤ < a b x a x b . are semi-closed or semi-open. The former is open at a and closed at b while the latter is closed at a and open at b. 1.3 BOUNDED AND UNBOUNDED SETS: SUPREMUM, INFIMUM A set S of real numbers is said to be bounded above if ∃ a real number K such that every member of S is less than or equal to K, i.e., , x K x S ≤ ∀ ∈ . The number K is called an upper bound of S. If no such number K exists, the set is said to be unbounded above or not bounded above. The set S is said to be bounded below if ∃ a real number k such that every member of S is greater than or equal to k, i.e., , k x x S ≤ ∀ ∈ . The number k is called a lower bound of S. If no such number k exists, the set is said to be unbounded below or not bounded below. A set is said to be bounded if it is bounded above as well as below. It may be seen that if a set has one upper bound, it has an infinite number of upper bounds. For if K is an upper bound of a set S then every number greater than K is also an upper bound of S. Thus every set S bounded above determines an infinite set—the set of its upper bounds. This set of upper bounds is bounded below in as much as every member of S is a lower bound thereof. Similarly, a set S bounded below determines an infinite set of its lower bounds, which is bounded above by the members of S. A member G of a set S is called the greatest member of S if every member of S is less than or equal to G, i.e., (i) G S ∈ (ii) , x G ≤ x S ∀ ∈ Similarly, a member g of the set is its smallest (or the least) member if every member of the set is greater than or equal to g. Clearly, a set may or may not have the greatest or the least member but an upper (lower) bound of the set, if it is a member of the set, is its greatest (least) member. A finite set always has the greatest as well as the smallest member. If the set of all upper bounds of a set S has the smallest member, say M, then M is called the least upper bound (l.u.b.) or the supremum of S.
- 26. Real Numbers 11 Clearly, the supremum of a set S may or may not exist and in case it exists, it may or may not belong to S. The fact that supremum M is the smallest of all the upper bounds of S may be described by the following two properties: (i) M is the upper bound of S, i.e., , . x M x S ≤ ∀ ∈ (ii) No number less than M can be an upper bound of S, i.e., for any positive number , ε however small, ∃ a number y S ∈ such that y M ε > − Again it may be seen that a set cannot have more than one supremum. For, let if possible, M and ′ M be two suprema of a set S, so that M and M ′ are both upper bounds of S. Also M is the l.u.b. and M ′ is an upper bound of S. ∴ M M ′ ≤ ...(1) Again M ′ is the l.u.b. and M is an upper bound of S. ∴ M M ′ ≤ ...(2) From (1) and (2) it follows that . M M ′ = If the set of all lower bounds of a set S has the greatest member say m then m is called the Greatest lower bound (g.l.b.) or the Infimum of S. Like the supremum, the infimum of a set may or may not exist and it may or may not belong to S. It can be easily shown that a set cannot have more than one infimum. The infimum m of a set S has the following two properties: (i) m is the lower bound of S, i.e. , . m x x S ≤ ∀ ∈ (ii) No number greater than m can be a lower bound of S, i.e., for any positive number ε, however small, ∃ a number z S ∈ such that . z m ε < + ILLUSTRATIONS 1. The set N of natural numbers is bounded below but not bounded above. 1 is a lower bound. 2. The sets I, Q and R are not bounded. 3. Every finite set of numbers is bounded. 4. The set S1 of all positive real numbers 1 { : 0, } S x x x = > ∈ R is not bounded above, but is bounded below. The infimum 0 is not a member of the set S1. 5. The infinite set 2 { : 0 1, } S x x x = < < ∈ R is bounded with supremum 1 and infimum 0, both of which do not belong to S2. 6. The infinite set 3 { : 0 1, } S x x x = ≤ ≤ ∈ Q is bounded, with supremum 1 and infimum 0 both of which are members of S3.
- 27. 12 Principles of Real Analysis 7. The set 4 1 : S n n = ∈ N is bounded. The supremum 1 belongs to S4 while infimum 0 does not. 8. Each of the following intervals is bounded: [a, b], ]a, b], [a, b[, ]a, b[. Example 1.3. Prove that the greatest member of a set, if it exists, is the supremum (l.u.b.) of the set. Solution. Let G be the greatest member of the set S. Clearly, , x G x S ≤ ∀ ∈ so that G is an upper bound of S. Again no number less than G can be an upper bound of S for if y be any number less than G, there exists at least one member g of S which is greater than y. Thus, G is the least of all the upper bounds of S, i.e., G is the supremum of S. EXERCISE 1. Give examples of sets which are (i) Bounded (ii) Not bounded (iii) Bounded below but not bounded above (iv) Bounded above but not bounded below. 2. Find the infimum and the supremum of the following sets. Which of these belongs to the set? (i) [1, 3, 5, 7, 9] (ii) 1 1 1 1, , , , ... 2 3 4 − − − − (iii) 1 ; n n ∈ N (iv) ( 1) ; n n n − ∈ N (v) 3 4 5 1 2, , , , ..., , ... 2 3 4 n n + − − − − − (vi) ( 1) 1 ; n n n − + ∈ N (vii) ]a, b[ (viii) [a, b[. 3. Which of the sets in question 2 are bounded? 4. Find the smallest and the greatest members (if they exist) for sets in question 2. 5. Show that the greatest (or the smallest) member of a set, in case it exists, is unique. 6. Show that the smallest member of a set, if it exists, is the infimum of the set. 7. Is the converse of the solved example 1.3, true? ANSWERS 2. (i) 1, 9; both (ii) –1, 0; infimum (iii) 0, 1; supremum (iv) 1 1, ; both 2 − (v) –2, –1; infimum (vi) 3 0, ; both 2 (vii) a, b; none (viii) a, b; infimum.
- 28. Real Numbers 13 3. All sets are bounded. 4. (i) 1, 9 (ii) –1, does not exist (iii) does not exist, 1 (iv) 1 1, 2 − (v) –2, does not exist (vi) 3 0, 2 (vii) do not exist (viii) a, does not exist. 1.4 COMPLETENESS IN THE SET OF REAL NUMBERS We have seen that all the properties—the properties of an ordered field, described so far, are possessed by the two sets, the set of real numbers R and the set of rational numbers Q. We shall now state a property, the property of completeness (or order-completeness) which is possessed by R and not by Q. This property not only distinguishes R from Q, but together with the ordered field property, it characterises R i.e., the set of real numbers is the only set which is a Complete Ordered Field. 1.4.1 Order-Completeness in R (O.C.) Every non-empty set of real numbers which is bounded above has the supremum (or the least upper bound) in R. In other words, the set of upper bounds of a non-empty set of real numbers bounded above has the smallest member. If S is a set of real numbers which is bounded above, then by considering the set { : } T x x S = − ∈ we may state the completeness property in the alternative form as: Every non-empty set of real numbers which is bounded below has the infimum (or g.l.b.) in R. Or, equivalently the set of lower bounds of a non-empty set of real numbers bounded below has the greatest member. We have thus completed the description of the set of real numbers as a Complete Ordered Field. We shall, however, show that the property of completeness does not hold good for the ordered field of rational numbers, i.e., the ordered field Q of rational is not order complete. Theorem 1.1. The set of rational numbers is not order-complete. To show that the set of rational numbers does not possess the property of completeness, it will suffice to show that there exists a non-empty set S of rationals (a subset of Q) which is bounded above but does not have a supremum in Q, i.e., no rational number exists which can be the supremum of S. Let S be the set (a subset of Q) of all those positive rational numbers whose square is less than 2, i.e., 2 { : , 0 2} S x x x x = ∈ > ∧ < Q Since 1 , S ∈ the set S is non-empty. Clearly 2 is an upper bound of S, therefore, S is bounded above. Thus, S is a non-empty set of rational numbers, bounded above. Let, if possible, the rational number K be its least upper bound. Clearly K is positive. Also by the law of trichotomy (0 – 1) which holds good in Q, one and only one of (i) K2 < 2, (ii) K2 = 2, (iii) K2 > 2 holds.
- 29. 14 Principles of Real Analysis (i) K2 < 2. Let us consider the positive rational number 4 3 3 2 K y K + = + Then ( ) 2 2 2 4 3 0 3 2 3 2 K K K y K K K − + − = − = < + + ⇒ y K > ...(1) Also, ( ) 2 2 2 2 4 3 2 – 2 2 0 3 2 3 2 K K y K K + − = − = > + + ⇒ 2 2 y y S < ⇒ ∈ ...(2) Thus, the member y of S is greater than K, so that K cannot be an upper bound of S and hence, there is a contradiction. (ii) K2 = 2. We have already shown that there exists no rational number whose square is equal to 2. Thus, this case is not possible. (iii) K2 > 2. Considering the positive rational number y as defined in case (i), we may easily deduce from (1) and (2) respectively that y < K and y2 > 2 Hence, there exists an upper bound y of S smaller than the least upper bound K, which is a contradiction. Thus, none of the three possible cases holds. Hence, our supposition that a rational number K is the least upper bound of S is wrong. Thus, no rational number exists which can be the least upper bound of S. Note. If we admit K in R and regard S as a set of real numbers then by the order completeness property, the supremum K of S exists in R. Clearly K > 0 and 2 2 2 2 Sup K y y K K S < ⇒ < ∧ > ⇒ ≠ 2 2 2 2 Sup K y y K K S > ⇒ > ∧ < ⇒ ≠ Thus by property 0–1, it follows that K2 = 2, i.e., the least upper bound K exists whose square is equal to 2. Further, since , K ∉ Q it follows that K is an irrational number. Similarly, it may be seen that there exist real numbers other than rational numbers whose squares are 2, 5, 7, ... etc. This establishes the existence of irrational numbers. Exercise. Show that the set of natural numbers is order-complete. 1.4.2 Archimedean Property of Real Numbers The order-completeness property has important consequences, one of which is the Archimedean property of real numbers which we now proceed to prove. Theorem 1.2. The real number field is Archimedean, i.e., if a and b be any two positive real numbers then there exists a positive integer n such that na > b.
- 30. Real Numbers 15 Let a, b be any two positive real numbers. Let us suppose, if possible, that for all positive integers ( ), . n na b + ∈ ≤ I Thus, the set { : } S na n + = ∈ I is bounded above, b being an upper bound. By the completeness property of the ordered-field of real numbers, set S must have the supremum M. ∴ , ≤ na M + ∀ ∈ n I ⇒ ( ) 1 , n a M + ≤ n + ∀ ∈ I ⇒ , na M a ≤ − n + ∀ ∈ I i.e., M – a is an upper bound of S. Thus a number, M – a less than the supremum M(l.u.b.) is an upper bound of S, which is a contradiction and hence our supposition is wrong. Hence the theorem. Corollary 1. If a be a positive real number and b any real number then there exists a positive integer n such that na > b. Corollary 2. For any positive real number a there exists a positive integer n such that n > a. The result follows by considering the two positive real numbers 1 and a. Corollary 3. If a be any real number then there exists a positive integer n such that n > a. For 0, a ≤ any positive integer n > a, and for a > 0, result follows by cor. 2. 1.4.3 Dedekind’s Form of Completeness Property We now state the completeness property of real numbers in another form, due to Dedekind, which states: If all the real numbers be divided into two non-empty classes L and U such that every member of L is less than every member of U, then there exists a unique real number, say , α such that every real number less than α belongs to L and every real number greater than α belongs to U. Clearly, the two classes L and U so defined are disjoint and the number α itself belongs either to L or U. The property of real numbers referred to above is known as Dedekind’s property. We may restate. Dedekind’s Property. If L and U are two subsets of R such that (i) , L U φ φ ≠ ≠ (each class has at least one member), (ii) L U = R 7 (every real number has a class) (iii) Every member of L is less than every member of U, i.e., x L y U x y ∈ ∧ ∈ ⇒ < Then either L has the greatest member or U has the smallest member. 1.4.4 Let us now prove the equivalence of the two forms of completeness. (a) First we show that the order completeness property of real numbers implies Dedekind’s property. The set R has the order completeness property i.e., every non-empty subset of R which is bounded above (below) has the Supremum (Infimum). Let L, U be two subsets of R such that
- 31. 16 Principles of Real Analysis (i) , ; L U φ φ ≠ ≠ (ii) ; L U = R 7 and (iii) Every member of L is less than every member of U. We have to show that either L has the greatest member or U has the smallest. By (iii) the non-empty set L is bounded above. If L has the greatest member, it establishes the result. If L has no greatest member, then by the order completeness property, the set of its upper bounds, which coincides with U, has the smallest member. This either L has the greatest member or U has the smallest member. (b) Let, now, R satisfy the Dedekind’s property. We shall show that R also satisfies the order completeness property. Let S be a non-empty set of real numbers bounded above, then we have to prove that S has the supremum. Let L and U be two sets of real numbers defined by L = {x: x is not an upper bound of S}, U = {x: x is an upper bound of S}. It may be easily seen that (i) , , L U φ φ ≠ ≠ (ii) , L U = R 7 and (iii) . x L y U x y ∈ ∧ ∈ ⇒ < Then by Dedekind property, either L has the greatest member or U has the smallest member. We shall show that L cannot have the greatest member. Let, if possible, L have the greatest member, say . ξ Then L ξ ξ ∈ ⇒ is not an upper bound of S. an such that . a S a ξ ⇒ ∃ ∈ < Now the real number 2 a ξ + is such that 2 a a ξ ξ + < < Since 2 a ξ + is greater than the greatest member ξ of L, ∴ 2 a U ξ + ∈ ⇒ 2 a ξ + is an upper bound of S. ...(1) Again, since 2 a ξ + is less than the member a of S,
- 32. Real Numbers 17 ∴ 2 a L ξ + ∈ ⇒ 2 a ξ + is not an upper bound of S. ...(2) Thus, we arrive at contradictory conclusions, and as such L has no greatest member. Thus, it follows that U, the set of upper bounds of S, has the smallest member, i.e., the set S has the supremum. We have thus proved the equivalence of Dedekind’s and the order completeness property of R. 1.4.5 Explicit Statement of the Properties of the Set of Real Numbers as a Complete-Ordered Field The set R of real numbers is a complete-ordered field because for arbitrary members, a, b, c of R, it satisfies the following conditions: A-1. , a b a b ∈ ⇒ + ∈ R R A-2. a + b = b + a A-3. (a + b) + c = a + (b + c) A-4. ∃ a member 0 in R such that a + 0 = a A-5. To each , a ∈ ∃ R an element a R − ∈ such that a + (–a) = 0 M-1. , a b ab ∈ ⇒ ∈ R R M-2. ab = ba M-3. (ab)c = a(bc) M-4. ∃ a member 1 in R such that 1 a a ⋅ = M-5. To each 0 , a ≠ ∈ ∃ R an element 1 a− ∈ R such that aa–1 = 1. AM. a(b + c) = ab + ac. O-1. For any two elements a, b of R, one and only one of the following is true: , , a b a b b a > = > O-2. a b b c a c > ∧ > ⇒ > O-3. a b a c b c > ⇒ + > + O-4. 0 a b c ac bc > ∧ > ⇒ > O.C. Every non-empty subset of R which is bounded above (below) has the supremum (infimum) in R. 1.4.6 Representation of Real Numbers as Points on a Straight Line Points on a line can be used to represent real numbers. This geometrical representation of real numbers is sometimes very useful and suggestive especially to the beginner. But this should not stop us from giving the proper proof of a theorem which may otherwise seem to be obvious.
- 33. 18 Principles of Real Analysis Let X'X be a straight line. Mark two points O and A on it such that A is to the right of O. The point O divides the line X'X into two parts; the part to the right of O, containing A, may be called positive and that to the left of O as negative. Such a line for which positive and negative sides are fixed is called a directed line. Let us consider the points O and A to represent rational numbers zero and 1 respectively, so that the distance OA is unity on a certain scale. To represent a rational number m/n (n > 0), take a point P on the right of O if m is positive and to the left of O if m is negative, such that OP in m times the nth part of the unit length OA. Of course the point P coincides with O if m is zero. The point P thus represents the rational number m/n. We may say that the rational number m/n corresponds to the point P or the point P corresponds to the rational number m/n. This way any rational number can be made to correspond to a point on the line. If points on the line corresponding to rational numbers be termed as rational points. We see that infinite number of rational points lie between any two different rational points, i.e., between any two rationals, there lie infinitely many rationals. Even though the rational points seem to cover a straight line very closely, there remain points on the line which are not rational. For example, the point Q on the line such that OQ is equal to the diagonal of the square with side OA does not correspond to any rational number. Also a point R such that OR is any rational multiple of OQ is also such a point. In fact there are infinitely many such points on the line. Hence, the set Q of rational numbers is insufficient to provide a complete picture of the straight line. Such points on the line which are not rational, and which may be supposed to fill up the gaps between rational points are called irrational points and these correspond to irrational numbers. In fact, there is at least one irrational between two rationals. Thus like rationals, there are infinitely many irrationals. Hence, every real number can be represented on the directed line and there seem to be as many points on the directed line as the real numbers. The same fact is expressed by Dedekind-Cantor Axiom which states: To every real number there corresponds a unique point on a directed line and to every point on a directed line there corresponds a unique real number. In view of the order completeness property, the set of real numbers R does not have gaps of the kind Q has, and thus forms a continuous system. On account of this characteristic, the set R is called the Arithmetical Continuum and the set of points on a line as the Geometrical Continuum. In view of the above axiom, we see that there is a one-one correspondence between the two continuum and accordingly we may use the word point for real number, and the real line for the directed line. It is evident that between any two real numbers, there exist infinitely many real numbers both rational and irrational. This is the property of denseness of the real number system. 1.5 ABSOLUTE VALUE OF A REAL NUMBER The absolute value, the numerical value or the modulus of a real number x, denoted by | x |, is defined as , if 0 , if 0 x x x x x ≥ = − < X O A Q P R m n ′ X
- 34. Real Numbers 19 Thus, we always have 0 x ≥ Also by definition x x − = Some theorems which are immediate consequences of the definition will now follow: Theorem 1.3. ( ) , x max x x = − . Now , if 0 x x x x = ≥ − ≥ Also , if 0 x x x x = − > < Thus in either case | x | is greater of the two numbers, x and –x, i.e., ( ) = max , − x x x . Corollary 1. ( ) ( ) = max – , − − − x x x ( ) max – , x x x = = ∴ x x − = Corollary 2. ( ) max , x x x x = − ≥ ∴ x x ≥ Theorem 1.4. - = - x x x min , > C Now , if 0 x x x x − = − < > Also ( ) – – , if 0 x x x x x − = − = < < Thus in either case –| x | is smaller of the two numbers x and –x, i.e., ( ) min , x x x − = − . Corollary. ( ) min , . x x x x − = − ≤ ∴ x x − ≤ Theorem 1.5. If , , x y ∈ R then (i) 2 2 2 x x x = = − (ii) xy x y = ⋅ (iii) , 0 = ≠ x x provided y y y .
- 35. 20 Principles of Real Analysis (i) For 0, x ≥ 2 2 x x x x = ⇒ = For x < 0. ( ) 2 2 2 x x x x x = − ⇒ = − = Thus in either case 2 2 x x = Similarly, ( ) 2 2 2 x x x − = − = Hence, 2 2 2 x x x = = − (ii) ( ) 2 2 2 2 2 2 2 ( ) xy xy x y x y x y = = = ⋅ = ⋅ ∴ xy x y = ± ⋅ But since | | xy and | | | | x y ⋅ are both non-negative, we take only the positive sign. ∴ xy x y = ⋅ (iii) 2 2 2 2 2 2 2 x x x x x y y y y y = = = = But since and x x y y are both non-negative, therefore taking positive square root of both sides, we have , when 0. x x y y y = ≠ Theorem 1.6. (Triangle inequalities). For all real numbers x, y show that (i) , x y x y and + ≤ + (ii) . x y x y − ≥ − (i) First Method: ( ) 2 2 2 2 2 x y x y x y xy + = + = + + 2 2 2 x y x y ≤ + + ⋅ xy xy x y ≤ = ⋅ 3 ( ) 2 x y = + Since | x + y | and | x | + | y | are both non-negative, therefore taking positive square roots of both sides, we have x y x y + ≤ +
- 36. Real Numbers 21 Second Method: When 0. x y + ≥ x y x y + = + x y ≤ + and x x y y ≤ ≤ 3 When x + y < 0, ( ) ( ) ( ) x y x y x y + = − + = − + − x y ≤ − + − and x x y y − ≤ − − ≤ − 3 But , x x y y − = − = Thus in either case, . x y x y + ≤ + (ii) First Method: 2 2 2 2 ( ) 2 x y x y x y xy − = − = + − 2 2 2 x y x y ≥ + − ⋅ ( ) xy xy x y − ≥− = − ⋅ 3 ( ) 2 2 x y x y = − = − Since, | | and | | | | x y x y − − are both non-negative, therefore taking the positive square root of both sides, we have . x y x y − ≥ − Second Method: Now ( ) x x y y x y y = − + ≤ − + [by part (i)] ∴ x y x y − ≥ − ...(1) Again ( ) . y y x x y x x = − + ≤ − + ∴ ( ) y x y x x y − ≥ − = − − But y x x y − = − ∴ ( ) x y x y − ≥ − − ...(2) From (1) and (2), ( ) { } max , x y x y x y − ≥ − − − x y = − Hence, x y x y − ≥ −
- 37. 22 Principles of Real Analysis Example 1.4. For real numbers , , 0 x a ε > show that (i) , x x ε ε ε < ⇔ − < < (ii) . x a a x a ε ε ε − < ⇔ − < < + Solution. (i) ( ) max , ε = − < x x x ε ε ⇔ < ∧ − < x x ε ε ⇔ < ∧ − < x x ε ε ⇔ − < < x (ii) ( ) ( ) { } max , x a x a x a ε − = − − − < ( ) ( ) x a x a ε ε ⇔ − < ∧ − − < x a a x ε ε ⇔ < + ∧ − < a x a ε ε ⇔ − < < + . Example 1.5. Show that for a bounded set S of real numbers there exists a number G > 0 such that , x G ≤ . x S ∀ ∈ Solution. Let the set S, with or without zero, contain both positive and negative numbers. Since it is bounded, it is bounded both above and below. Let k < 0 be a lower bound and K > 0 be an upper bound. Now , ∀ ∈ x S | | ≤ ⇒ ≤ x K x K when x > 0 and | | ≤ ⇒ ≤ x k x k when x < 0 So that , | | max (| |, | |). ∀ ∈ ≤ x S x K k Thus on taking a real number G = max (|K|, |k| + 1), we have , . < ∀ ∈ x G x S EXERCISE 1. Show that (i) . x x ≥ − (ii) . x y y x − = − (iii) . x y x y − ≤ + 2. If x1, x2, x3, ..., xn are real numbers, then show that (i) 1 2 1 2 ... ... . n n x x x x x x + + + ≤ + + + (ii) 1 2 1 2 ... ... . n n x x x x x x = ⋅
- 38. CHAPTER 2 Limit Points: Open and Closed Sets 2.1 INTRODUCTION In this chapter, we shall study the concept of neighbourhood of a point, open and closed sets, and limit points of a set of real numbers and the Bolzano-Weierstrass theorem, which is one of the most fundamental theorems of Real Analysis and lays down a sufficient condition for the existence of limit points of a set. We shall be dealing only with real numbers and sets of real numbers unless otherwise stated. 2.1.1 Neighbourhood of a Point A set N ⊂ R is called the neighbourhood of a point a, if there exists an open interval I containing a and contained in N, i.e., a I N ∈ ⊂ It follows from the definition that an open interval is a neighbourhood of each of its points. Though open intervals containing the point are not the only neighbourhoods of the point but they prove quite adequate for a discussion like ours and are more expressive of the idea of neighbourhoods as understood in ordinary language. We shall, therefore, whenever convenient, take the open interval ] – , [ a a δ δ + where 0 δ > as a neighbourhood of the point a. Deleted Neighbourhoods The set { : 0 | | }, x x a δ < − < i.e., an open interval ] – , [ a a δ δ + from which the number a itself has been excluded or deleted is called a deleted neighbourhood of a. Note. For the sake of brevity, we shall write neighbourhood as ‘nbd’. ILLUSTRATIONS 1. The set R of real numbers is the neighbourhood of each of its points. 2. The set Q of rationals is not the nbd of any of its points. 3. The open interval ]a, b[ is nbd of each of its points. 4. The closed interval [a, b] is the nbd of each point of ]a, b[ but is not a nbd of the end points a and b.
- 39. 24 Principles of Real Analysis 5. The null set φ is a nbd of each of its points in the sense that there is no point in φ of which it is not a nbd. Example 2.1. A non-empty finite set is not a nbd of any point. Solution. A set can be a nbd of a point if it contains an open interval containing the point. Since an interval necessarily contains an infinite number of points, therefore, in order that a set be a nbd of a point it must necessarily contain an infinity of points. Thus a finite set cannot be a nbd of any point. Example 2.2. Superset of a nbd of a point x is also a nbd of x. i.e., if N is a nbd of a point x and M N ⊃ , then M is also a nbd of x. Example 2.3. Union (finite or arbitrary) of nbds of a point x is again a nbd of x. Example 2.4. If M and N are nbds of a point x, then show that M N ∩ is also a nbd of x. Solution. Since M, N are nbds of x, ∃ open intervals enclosing the points x such that ] [ ] [ 1 1 2 2 – , and – , x x x M x x x N δ δ δ δ ∈ + ⊂ ∈ + ⊂ Let ( ) 1 2 min , . δ δ δ = Then ] [ ] [ 1 1 – , , x x x x M δ δ δ δ + ⊂ − + ⊂ and ] [ ] [ 2 2 – , – , x x x x N δ δ δ δ + ⊂ + ⊂ ⇒ ] [ – , x x M N δ δ + ⊂ ∩ ⇒ M N ∩ is a nbd of x. 2.1.2 Interior Points of a Set A point x is an interior point of a set S if S is a nbd of x. In other words, x is an interior point of S if ∃ an open interval ]a, b[ containing x and contained in S, i.e., ] , [ . x a b S ∈ ⊂ Thus a set is a neighbourhood of each of its interior points. Interior of a Set. The set of all interior points of a set is called the interior of the set. The interior of a set S is generally denoted by Si . Exercise 1. Show that the interior of the set N or I or Q is the null set, but interior of R is R. Exercise 2. Show that the interior of a set S is a subset of S, i.e., . i S S ⊂ 2.1.3 Open Set A set S is said to be open if it is a nbd of each of its points, i.e., for each , x S ∈ there exists an open interval x I such that . x x I S ∈ ⊂ Thus every point of an open set is an interior point, so that for an open set S, Si = S. Evidently, S is open i S S ⇔ =
- 40. Limit Points: Open and Closed Sets 25 Of course the set is not open if it is not a nbd of at least one of its points or that there is at least one point of the set which is not an interior point. ILLUSTRATIONS 1. The set R of real numbers is an open set. 2. The set Q of rationals is not an open set. 3. The closed interval [a, b], is not open for it is not a neighbourhood of the end points a and b. 4. The null set φ is open, for there is no point in φ of which it is not a neighbourhood. 5. A non-empty finite set is not open. 6. The set 1 : ∈ n n N is not open. Exercise. Give an example of an open set which is not an interval. Example 2.5. Show that every open interval is an open set. Or, every open interval is a nbd of each of its points. Solution. Let x be any point of the given open interval ]a, b[ so that we have a < x < b. Let c, d be two numbers such that a < c < x, and x < d < b so that we have ] [ ] [ , , . a c x d b x c d a b < < < < ⇒ ∈ ⊂ Thus the given interval ]a, b[ contains an open interval containing the point x, and is therefore a nbd of x. Hence, the open interval is a nbd of each of its points and is therefore an open set. Exercise. Show that every point of an open interval is its interior point. Example 2.6. Show that every open set is a union of open intervals. Solution. Let S be an open set and xλ a point of S. Since S is open, therefore ∃ an open interval x I λ for each of its points xλ such that x x I S x S λ λ λ ∈ ⊂ ∀ ∈ Again the set S can be thought of as the union of singleton sets like { }, xλ i.e., { }, S xλ λ∈Λ = U where Λ is the index set ∴ { } x S x I S λ λ λ λ ∈Λ ∈Λ = ⊂ ⊂ U U ⇒ λ λ∈Λ = U x S I a c x d b
- 41. 26 Principles of Real Analysis Theorem 2.1. The interior of a set is an open set. Let S be a given set, and i S its interior. If , i S φ = then i S is open. When , i S φ ≠ let x be any point of . i S As x is an interior point of S, ∃ an open interval x I such that . x x I S ∈ ⊂ But , x I being an open interval, is a nbd of each of its points. ⇒ every point of x I is an interior point of , x I and x I S ⊂ ⇒ every point of x I is interior point of S. ∴ i x I S ⊂ ⇒ i x x I S ⊂ ⊂ ⇒ any point x of i S is interior point of i S ⇒ i S is an open set. Corollary. The interior of a set S is an open subset of S. Theorem 2.2. The interior of a set S is the largest open subset of S. or The interior of a set S contains every open subset of S. We know that the interior Si of a set S is an open subset of S. Let us now proceed to show that any open subset S1 of S is contained in . i S Let x be any point of S1. Since an open set is a nbd of each of its points, therefore S1 is a nbd of x. But S is a superset of S. ∴ S is also a nbd of x ⇒ x is an interior point of S ⇒ i x S ∈ Thus 1 i x S x S ∈ ⇒ ∈ ∴ 1 i S S ⊂ Hence, every open subset of S is contained in its interior . i S ⇒ , i S the interior of S, is the largest open subset of S. Corollary. Interior of a set S is the union of all open subsets of S. Theorem 2.3. The union of an arbitrary family of open sets is open. Let F be the union of an arbitrary family { : } Sλ λ = ∈ Λ F of open sets, Λ being an index set. To prove that F is open, we shall show that for any point , x F ∈ it contains an open interval containing x. Let x be any point of F. Since F is the union of the members of , ∃ F at least one member, say Sλ of F which contains x. Again, Sλ being an open set, ∃ an open interval x I such that
- 42. Limit Points: Open and Closed Sets 27 . x x I S F λ ∈ ⊂ ⊂ Thus the set F contains an open interval containing any point x of F F ⇒ is an open set. Theorem 2.4. The intersection of any finite number of open sets is open. Let us consider two open sets S, T. If , S T φ ∩ = it is an open set. If , S T φ ∩ ≠ let x be any point of . S T ∩ Now x S T x S x T ∈ ∩ ⇒ ∈ Λ ∈ . ⇒ S, T are nbds of x. [ ] , are open S T Q ⇒ S T ∩ is a nbd of x. But since x is any point of , S T ∩ therefore S T ∩ is a nbd of each of its points. Hence, S T ∩ is open. The proof may of course be extended to a finite number of sets. Note. The above theorem does not hold for the intersection of arbitrary family of open sets. Consider for example the open sets 1 1 – , , n S n n n = ∈ N Their intersection is the set {0} consisting of the single point 0, and this set is not open. 2.2 LIMIT POINTS OF A SET Definition 1. A real number ξ is a limit point of a set ( ) S ⊂ R if every nbd of ξ contains an infinite number of members of S. Thus ξ is a limit point of a set S if for any nbd N of , ξ N S ∩ is an infinite set. A limit point is also called a cluster point, a condensation point or an accumulation point. A limit point of a set may or may not be a member of the set. Further it is clear from the definition that a finite set cannot have a limit point. Also it is not neccesary that an infinite set must possess a limit point. In fact a set may have no limit point, a unique limit point, a finite or an infinite number of limit points. A sufficient condition for the existence of a limit point is provided by Bolzano-Weierstrass theorem which is discussed in the next section. The following is another definition of a limit point. Definition 2. A real number ξ is a limit point of a set ( ) S ⊂ R if every nbd of ξ contains at least one member of S other than . ξ The essential idea here is that the points of S different from ξ get ‘arbitrarily close’ to ξ or ‘pile up’ at . ξ Evidently definition 1 implies definition 2. Let us now prove that definition 2 implies definition 1. x2 x1 ξ δ − 1 ξ ξ δ + 1
- 43. 28 Principles of Real Analysis Let ξ be a limit point of the set ( ) S ⊂ R such that every nbd of ξ contains at least one point of S other than . ξ Let 1 1 ] – , [ ξ δ ξ δ + be one such nbd of ξ which contains at least one point, say, 1 of . x S ξ ≠ Let 1 2 1 | | . x ξ δ δ − = < Now consider the nbd 2 2 ] – , [ ξ δ ξ δ + of ξ which by def. 2 of a limit point, must have one point, say, x2 of S other than . ξ By repeating the argument with the nbd 3 3 ] – , [ ξ δ ξ δ + of ξ where 3 2 | – | x δ ξ = and so on, it follows that the nbd ] – , [ of i i ξ δ ξ δ ξ + contains an infinity of members of S. Hence, Def. 2 ⇒ Def. 1. It is instructive to note that a point ξ is not a limit point of a set S if ∃ even one nbd of ξ not containing any point of S other than . ξ Exercise. Give a bounded set having (i) no limit point, (ii) infinite numbers of limit points. Derived Sets. The set of all limit points of a set S is called the derived set of S and is denoted by . ' S ILLUSTRATIONS 1. The set I has no limit point, for a nbd 1 1 2 2 – , m m + of , m ∈ I contains no point of I other than m. Thus the derived set of I is the null set . φ 2. Every point of R is a limit point, for, every nbd of any of its points contains an infinity of members of R. Therefore . ′ = R R 3. Every point of the set Q of rationals is a limit point, for, between any two rationals there exist an infinity of rationals. Further every irrational number is also a limit point of Q for between any two irrationals there are infinitely many rationals. Thus every real number is a limit point of Q, so that ′ = Q R. 4. The set 1 : n n ∈ N has only one limit point, zero, which is not a member of the set. 5. Every point of the closed interval [a, b] is its limit point, and a point not belonging to the interval is not a limit point. Thus the derived set [ , ] [ , ]. a b a b ′ = 6. Every point of the open interval ]a, b[ is its limit point. The end points a, b which are not members of ]a, b[ are also its limit points. Thus ] [ [ ] , , . a b a b ′ = Examples. Obtain the derived sets: 1. { } : 0 1 , x x ≤ < 2. { } : 0 1, , x x x < < ∈ Q 3. { } 1 1 1 2 2 3 1, – 1, 1 , – 1 , – 1 , , K 4. 1 1 : , n n + ∈ N
- 44. Limit Points: Open and Closed Sets 29 5. 1 1 : , . m n m n + ∈ ∈ N N 2.2.1 A finite set has no limit point. Also we have seen that an infinite set may or may not have limit points. We shall now discuss a theorem which sets out sufficient conditions for a set to have limit points. Bolzano-Weierstrass Theorem (for sets). Every infinite bounded set has a limit point. Let S be any infinite bounded set and m, M its infimum and supremum respectively. Let P be a set of real numbers defined as follows: x P ∈ iff it exceeds at the most a finite number of members of S. The set P is non-empty, for . ∈ M P Also M is an upper bound of P, for no number greater than or equal to M can belong to P. Thus the set P is non-empty and is bounded above. Therefore, by the order- completeness property, P has the supremum, say . ξ We shall now show that ξ is a limit point of S. Consider any nbd. ] – , [ ξ ε ξ ε + of , ξ where 0. ε > Since ξ is the supremum of P, ∃ at least one member say η of P such that – . η ξ ε > Now η belongs to P, therefore it exceeds at the most a finite number of members of S, and consequently – ( ) ξ ε η < can exceed at the most a finite number of members of S. Again as ξ is the supremum of P, ξ ε + cannot belong to P, and consequently ξ ε + must exceed an infinite number of members of S. Now – ξ ε exceeds at the most a finite number of members of S and ξ ε + exceeds infinitely many members of S. ] – , [ ξ ε ξ ε ⇒ + contains an infinite number of members of S. Consequently ξ is a limit point of S. 2.2.2 Example 2.7. If S and T are subsets of real numbers, then show that (i) , S T S T ′ ′ ⊂ ⇒ ⊂ and (ii) ( ) S T S T ′ ′ ′ ∪ = ∪ . Solution. (i) If S φ ′ = , then evidently . S T ′ ′ ⊂ When , S φ ′ ≠ let S ξ ′ ∈ and N be any nbd of . ξ ⇒ N contains an infinite number of members of S. But , S T ⊂ N ∴ contains infinitely many members of T ⇒ ξ is limit point of T, i.e., . T ξ ′ ∈ Thus . S T ξ ξ ′ ′ ∈ ⇒ ∈ Hence . S T ′ ′ ⊂ (ii) Now ( ) S S T S S T ′ ′ ⊂ ⇒ ⊂ U U and ( ) T S T T S T ′ ′ ⊂ ∪ ⇒ ⊂ ∪ Consequently, ( ) S T S T ′ ′ ′ ∪ ⊂ ∪ ...(1)
- 45. 30 Principles of Real Analysis Now we proceed to show that ( ) . S T S T ′ ′ ′ ∪ ⊂ ∪ If ( ) , S T φ ′ ∪ = then evidently ( ) . S T S T ′ ′ ′ ∪ ⊂ ∪ When ( ) , S T φ ′ ∪ ≠ let ( ) . S T ξ ′ ∈ ∪ Now ξ is a limit point of ( ), S T ∪ therefore, every nbd of ξ contains an infinite number of points of ( ) S T ∪ ⇒ every nbd of ξ contains infinitely many points of S or T or both. ⇒ ξ is a limit point of S or a limit point of T ⇒ . S T S T ξ ξ ξ ′ ′ ′ ′ ∈ ∨ ∈ ⇒ ∈ ∪ Thus ( ) . S T S T ξ ξ ′ ′ ′ ∈ ∪ ⇒ ∈ ∪ Consequently, ( ) S T S T ′ ′ ′ ∪ ⊂ ∪ ...(2) From (1) and (2) it follows that ( ) S T S T ′ ′ ′ ∪ = ∪ Thus the derived set of the union = the union of the derived sets. (ii) Aliter. To show that ( ) S T S T ′ ′ ′ ∪ ⊂ ∪ We may show that ( ) . S T S T ξ ξ ′ ′ ′ ∉ ∪ ⇒ ∉ ∪ Now S T ξ ′ ′ ∉ ∪ implies that ξ does not belong to either. ⇒ ξ is not a limit point of S or of T ∴∃ nbds. 1 2 , of N N ξ such that N1 contains no point of S other than ξ and N2 contains no point of T other than possibly . ξ Again, since 1 2 1 1 2 2 , N N N N N N ∩ ⊂ ∩ ⊂ therefore ∃ a nbd. 1 2 of N N ξ ∩ which contains no point other than ξ of S or of T and thus of S T ∪ . ⇒ ξ is not a limit point S T ∪ . ⇒ ( )′ ∉ ∪ S T ξ Thus ( ) S T S T ξ ξ ′ ′ ′ ∉ ∪ ⇒ ∉ ∪ so that ( ) S T S T ′ ′ ′ ∪ ⊂ ∪ Example 2.8. (i) If S, T are subsets of R, then show that ( ) S T S T ′ ′ ′ ∩ ⊂ ∩ . (ii) Give an example to show that ( ) S T ′ ∩ and S T ′ ′ ∩ may not be equal. Solution. (i) Now ( ) S T S S T S ′ ′ ∩ ⊂ ⇒ ∩ ⊂ and ( ) S T T S T T ′ ′ ∩ ⊂ ⇒ ∩ ⊂ Consequently, ( ) S T S T ′ ′ ′ ∩ ⊂ ∩ (ii) Let S = ]1, 2[ and T = ]2, 3[, so that ( ) . S T S T φ φ φ ′ ′ ∩ = ⇒ ∩ = =
- 46. Limit Points: Open and Closed Sets 31 Also [ ] [ ] 1, 2 , 2, 3 S T ′ ′ = = ∴ { } 2 . S T ′ ′ ∩ = Thus ( ) . S T S T ′ ′ ′ ∩ ≠ ∩ 2.3 CLOSED SETS: CLOSURE OF A SET 2.3.1 A real number ξ is said to be an adherent point of a set ( ) S ⊂ R if every nbd of ξ contains at least one point of S. Evidently an adherent point may or may not belong to the set and it may or may not be a limit point of the set. It follows from the definition that a number S ξ ∈ is automatically an adherent point of the set, for, every nbd of a member of the set contains atleast one member of the set, namely the member itself. Further a number S ξ ∉ is an adherent point of S only if ξ is a limit point of S, for, every nbd of ξ then contains atleast one point of S which is other than . ξ Thus the set of adherent points of S consists of S and the derived set . S′ The set of all adherent point of S, called the closure of S, is denoted by , S % and is such that S S S′ = ∪ % . ILLUSTRATIONS 1. . φ ′ = ∪ = ∪ = I I I I I % 2. . ′ = ∪ = ∪ = Q Q Q Q R R % 3. ′ = ∪ = ∪ = R R R R R R % 4. . φ φ φ φ φ φ ′ = ∪ = ∪ = % 2.3.2 Closed Sets A set is said to be closed if each of its limit points is a member of the set. In other words a set S is closed if no limit point of S exists which is not contained in S. In rough terms, a set is closed if its points do not get arbitrarily close to any point outside of it. Thus a set S is closed iff or . S S S S ′ ⊂ = % Consequently, a closed set is also defined as a set S for which . S S = % It should be clearly understood that the concept of closed and open sets are neither mutually exclusive nor exhaustive. The word not closed should not be considered equivalent to open. Sets exist which are both open and closed, or which are neither open nor closed. The set consisting of points of ]a, b] is neither open nor closed. ILLUSTRATIONS 1. [a, b] is a set which is closed but not open.
- 47. 32 Principles of Real Analysis 2. The set [0, 1] [2, 3], ∪ which is not an interval, is closed. 3. The null set φ is closed for there exists no limit point of φ which is not contained in . φ As was shown earlier, φ is also open. 4. The set R of real numbers is open as well as closed. 5. The set Q is not closed, for . ′ = ⊂ Q R | Q Also it is not open. 6. 1 : n n ∈ N is not closed, for it has one limit point, 0, which is not a member of the set. Also it is not open. 7. Every finite set A is a closed set, for its derived set . A A φ ′ = ⊂ 8. A set A which has no limit point coincides with its closure, for A φ ′ = and . A A A A ′ = ∪ = % 2.3.3 Typical Examples Example 2.9. Show that the set { : 0 1, } S x x x = < < ∈ R is open but not closed. Solution. The set S is the open interval ]0, 1[. ∴ It contains a nbd of each of its points. Hence, it is an open set. Again every point of S is a limit point. The end points 0 and 1 which are not members of the set are also limit points. Thus S is not closed. Example 2.10. Show that the set 1 1 1 1 1, 1, , , , – , ... 2 2 3 3 = − − S is neither open nor closed. Solution. The members of S heap or cluster near zero on both sides of it and every nbd of zero contains an infinite number of points of S. Thus 0 S ∉ is a limit point ⇒ S is not closed. Again S is not open for it does not contain any nbd of any of its points. For example, a nbd 1 1 1 1 , 3 100 3 100 − + of 1 3 is not contained in the set. Hence, the set is not open. Example 2.11. Show that the set 1 1 1 1 1, 1,1 , 1 ,1 , 1 , ... 2 2 3 3 − − − is closed but not open. Solution. 1 and –1 are the only limit points of the set and are in the set. Therefore, the set is closed. Again all members of the set (except 1, –1) are not the interior points of the set. Thus the set is not open. Hence, the set is closed but not open. The relationship between closed and open sets is brought out by Theorem 2.5 that follows and is sometimes taken as the definition of a closed set.
- 48. Limit Points: Open and Closed Sets 33 2.3.4 Some Important Theorems Theorem 2.5. A set is closed iff its complement is open. Necessary. Let S be a closed set. We shall show that its complement R – S = T is open. Let x be any point of T. . x T x S ∈ ⇒ ∉ Also, since S is closed, x cannot be a limit point of S. Thus ∃ a nbd N of x such that . N S φ ∩ = N T ⇒ ⊂ ⇒ every point of T is an interior point. Thus T is an open set. Sufficient. Let S be a set whose complement R – S = T is open. To show that S is closed, we shall show that every limit point of S is in S. Let, if possible, a limit point ξ of S be not in S so that ξ is in T. As T is open, ∃ a nbd of ξ contained in T and thus containing no point of S. ∴ ∃ a nbd N of ξ which contains no point of S. ⇒ ξ is not a limit point of S; which is a contradiction. Hence no limit point of S exists which is not in S. ∴ S is closed. Theorem 2.6. The intersection of an arbitrary family of closed sets is closed. Let F be the intersection set of an arbitrary family = { : } Sλ λ ∈ Λ F of closed sets, Λ being an index set. If the derived set F′ of F is , φ i.e., when F is a finite set or an infinite set without limit points, then evidently it is closed. When , F φ ′ ≠ let , F ξ ′ ∈ i.e. ξ be a limit point of F, so that every nbd of ξ contains infinitely many members of F and as such of each member Sλ of the familyF of closed sets. ⇒ ξ is limit point of each closed set , Sλ ⇒ ξ belongs to each . S S F λ λ λ ξ ∈ Λ ⇒ ∈ ∩ = Thus the set F is closed. Note. We have given an independent proof but on taking complements, this theorem follows from theorem 2.3. Theorem 2.7. The union of two closed sets is a closed set. Let S, T be the two given closed sets and ξ a limit point of F, where . F S T = ∪ We have to show that , F ξ ∈ for then, the set F will be closed. Let if possible , F ξ ∉ thus . S T ξ ξ ∉ Λ ∉ Also as S, T are closed sets, the point ξ which does not belong to them, cannot be a limit point of either.
- 49. 34 Principles of Real Analysis ∴ ∃ nbds N1 and N2 of ξ such that 1 2 . N S N T φ φ ∩ = ∧ ∩ = ...(1) Let 1 2 , where . N N N N ξ ∩ = ∈ ∴ From (1) it follows that ( ) . N S T N F φ φ ∩ ∪ = ⇒ ∩ = Thus ∃ a nbd N of ξ which contains no point of F. ⇒ ξ is not a limit point of F, which is a contradiction. Hence, no point not belonging to F can be its limit point, and consequently F S T = ∪ is a closed set. Remarks 1. The theorem can be extended to the union of a finite number of sets. So we may restate the theorem as: The union of a finite number of closed sets is closed. 2. We have given an independent proof but the theorem follows from theorem 4 on taking complements. 3. The union of an arbitrary family of closed sets may not always be a closed set. For example, let 1 , 2 , for . n S a a n a n = + + ∈ Λ ∈ N R Then ] ] , 2 , n n S a a ∈ ∪ = + N which is not a closed set. Theorem 2.8. The derived set of a set is closed. Let S′ be the derived set of a set S. We have to show that the derived set S′′ of S′ is contained in . S′ Now if , S φ ′′ = i.e., when S′ is either a finite set or an infinite set without limit points, then S S φ ′′ ′ = ⊂ and therefore S′ is closed. When , S φ ′′ ≠ let , S ξ ′′ ∈ i.e., ξ be a limit point of . S′ ∴ Every nbd N of ξ contains at least one point of . S η ξ ′ ≠ Again, S η η ′ ∈ ⇒ is a limit point of S ⇒ every nbd of , η N being such a nbd, contains infinitely many points of S. Thus every nbd N (of ) ξ contains an infinitely many points of S ⇒ ξ is a limit point of S, i.e., . S ξ ′ ∈ Consequently S S ξ ξ ′′ ′ ∈ ⇒ ∈ ∴ , S S ⊂ ′′ ′ i.e., S′ is a closed set. Corollary 1. S′′ is closed, and therefore the closure of S′ is , S′′ i.e., . S S′ = % Corollary 2. For every set S the closure S % is closed. We have simply to show that ( ) . S S ′ ⊂ % % Now, ( ) ( ) . S S S S S S S ′ ′ ′ ′ ′′ ′ = ∪ = ∪ = ⊂ % % (ref. § 2.2)
- 50. Limit Points: Open and Closed Sets 35 Corollary 3. For every set , S S S = % % % ( ) . S S S S ′ = ∪ = % % % % % Theorem 2.9. The supremum (infimum) of a bounded non-empty set ( ), ⊂ S R when not a member of S, is a limit point of S. Let M be the supremum of the bounded set ( ), S ⊂ R which must exist by the order completeness property of R. If , M S ∉ then for any number 0, ε > however small, ∃ at least one member x of S such that . M x M ε − < < Thus every nbd of M contains atleast one member x of the set S other than M. Hence M is a limit point of S. Corollary. The supremum (infimum) M of a bounded set S is always a member of the closure S % of S. When , M S ∈ M S M S S S ′ ∈ ⇒ ∈ ∪ = % When , M S ∉ M S M S M S S S ′ ′ ∉ ⇒ ∈ ⇒ ∈ ∪ = % Consequently . M S ∈ % Theorem 2.10. The derived set of a bounded set is bounded. Let m, M be the bounds of a set S. It will now be shown that no limit point of S can be outside the interval [m, M]. Let, if possible, M ξ > be a limit point of S, and ε be a positive number such that . M ε ξ < − Then since M is an upper bound of S, no member of S can lie in the interval ] , [, ξ ε ξ ε − + therefore ∃ a nbd of ξ which contains no point of S so that ξ cannot be a limit point of S. Hence S has no limit point greater than M. Similarly it can be shown that no limit point of S is less than m. Hence [ ] , . S m M ′ ⊂ Corollary. If S is bounded then so is its closure S %. [ ] [ ] [ ] , , , . S m M S m M S S S m M ′ ′ ⊂ ⇒ ⊂ ⇒ = ∪ ⊂ % Remark. If supremum M (infimum m) of S is not a member of S, then it is a limit point of S and in view of the above theorem, it is the greatest (least) member of . S′ However, if it is a member of S, then it is not necessarily a limit point of S, so that M (or m) may not be a member of [ , ]. S m M ′ ⊂ Thus M, m may not always be supremum and infimum of S′ but they are always so for . S S S′ = ∪ % For example, for the set 1 1 1 1 1,1, 1 ,1 , 1 , 1 , ... 2 2 3 3 = − − − S m M ξ ε − ξ ε + N
- 51. 36 Principles of Real Analysis { } 1 1 1 , 1 , 1,1 2 2 = − = = − ′ m M S inf 1 , Sup 1 S m S M ′ ′ = − ≠ = ≠ but inf , Sup . S m S M = = % % Theorem 2.11. The derived set S′ of a bounded infinite set ( ) S ⊂ R has the smallest and the greatest members. Since the set S is bounded, therefore S′ is also bounded. Also S′ is non-empty, for by Bolzano- Weierstrass theorem S has at least one limit point. Now S′ may be finite or inifinite. When ( ) S φ ′ ≠ is finite, evidently it has the greatest and the least members. When S′ is infinite, being bounded set of real numbers, by order-completeness property of R, it has the supremum G and the infimum g. It will now be shown that G, g are limit points of S, i.e., , G S g S ′ ′ ∈ ∈ Let us first consider G. ] , [, > 0, G G ε ε ε − + be any nbd of G. Now G being the supremum of , S′ ∃ at least one member ξ of S′ such that . G G ε ξ − < ≤ Thus ] , [ G G ε ε − + is a nbd of . ξ But ξ is a limit point of S, so that ] , [ G G ε ε − + contains an infinity of members of S ⇒ any ] , [ nbd G G ε ε − + of G contains an infinite number of members of S ⇒ G is a limit point of S G S′ ⇒ ∈ Similarly, it can be shown that g S ∈ ′. Thus G S g S ∈ ′ ∈ ′ and , being supremum and infimum of ′ S , are the greatest and the smallest members of ′ S . The theorem may be restated as: Every bounded infinite set has the smallest and the greatest limit points. The smallest and greatest limit points of a set are called the lower and upper limits of indetermination or simply the lower and upper limits respectively of the set.
- 52. Real Sequences 37 CHAPTER 3 Real Sequences In this chapter we shall study a special class of functions whose domain is the set N of natural numbers and range a set of real numbers—the Real Sequences. 3.1 FUNCTIONS Let A and B be two sets, and let there be a rule which associates to each member x of A, a member y of B. Such a rule or a correspondence f under which to each element x of the set A, there corresponds exactly one element y of the set B, is called a mapping or a function. Symbolically, we write : , f A B → i.e., f is a mapping or a function of A into B. The members of A are the elements to which some elements of B are to correspond. The set A is called the Domain of the function f. The set B contains all the elements which correspond to the elements of A, and is called to co-domain of f. The unique element of B which corresponds to an element x of A is called the image of x or the value of the function at x and is denoted by ( ); f x x is called the preimage of ( ). f x It may be observed that while every element of the domain finds its image of B, there may be some elements in B which are not the image of any element of the domain A. The set of all those elements of the co-domain B which are the images of the elements of the domain A is called the range or the range set of the function f. If f be a function with domain D and range , R′ then we say that f is a function from D onto . R′ In this book we shall be concerned only with the functions which are many to one, i.e. functions with respect to which one or more elements of the domain correspond to the same element of the range. If members of the domain set are denoted by x and those of the range set by y, then ( ); y f x = where ( ) f x is the value of the function at x. However in the classical presentation of the subject instead of referring to f as a function we generally refer to ( ) f x as function. In modern presentation of the subject such an erroneous terminology is being given up. 3.2 SEQUENCES A function whose domain is the set N of natural numbers and range a set of real numbers R is called a real sequence. Thus a real sequence is denoted symbolically as S: . N R → Since we shall be dealing with real sequences only, we shall use the term sequence to denote a Real sequence.
- 53. 38 Principles of Real Analysis NOTATION. Since the domain for a sequence is always N, a sequence is specified by the values S n n , . ∈N Thus a sequence may be denoted as S n n l q, ∈N or {S1, S2, S3, ... Sn, ...} The values S1, S2, S3, ... are called the first, second, ... terms of the sequence. The mth and nth terms Sm and Sn for m n ≠ are treated as distinct terms even if Sm = Sn. Thus the terms of a sequence are arranged in a definite order as first, second, third, ... terms and the terms occurring at different positions are treated as distinct terms even if they have the same value. The number of terms in a sequence is always infinite. In other words, we define a sequence as an ordered set of real numbers whose members can be put in a one-one correspondence with the set of natural numbers. However, a sequence may have only a finite number of distinct elements. For example, 1. S n n n l q b g { } = − ∈ 1 , . N Here S1 = –1 , S2 = 1, S3 = –1, S4 = 1, ... so that there are only two, 1, –1 distinct elements. 2. S n n n l q= R S T U V W ∈ 1 , . N Here S1 = 1, 2 3 1 1 , ,... 2 3 = = S S All the elements are distinct. ILLUSTRATIONS 1. {Sn}, where S n n n n = + F HG I KJ ∈ 1 1 , . N 2. {Sn}, where S n n n = + − ∈ 1 1 b g , . N 3. {Sn}, where S n n = ∀ ∈ 1, . N 4. ( ) 1 1 , . ! − − ∈ n n n N 3.2.1 The Range The Range or the Range Set is the set consisting of all distinct elements of a sequence, without repetition and without regard to the position of a term. Thus the range may be a finite or an infinite set, without ever being the null set. 3.2.2 Bounds of a Sequence Bounded above sequences A sequence {Sn} is said to be bounded above if there exists a real number K such that
- 54. Real Sequences 39 S K n ≤ ∀ ∈ n N Bounded below sequences A sequence {Sn} is said to be bounded below if there exists a real number k such that S k n n ≥ ∀ ∈N. Bounded sequences A sequence is said to be bounded when it is bounded both above and below. K and k are respectively the upper and the lower bounds of the sequence. Evidently a sequence is bounded iff its range is bounded. Also the bounds of the range are the bounds of the sequence. 3.2.3 Convergence of Sequences Definition 1. A sequence {Sn} is said to converge to a real number l (or to have the real number l as its limit) if for each A > 0, there exists a positive integer m (depending on ε ) such that , n S l ε − < for all n m ≥ . The fact is expressed by saying that the terms approach the value l or tend to l as n becomes larger and larger. The same thing expressed in symbols is S l n n ® ® ¥ for or lim . n n S l ®¥ = The definition ensures that (i) From some stage onwards the difference between Sn and l can be made less than any preassigned positive number ε, however small, i.e., given any positive real number ε, no matter however small, ∃ a positive integer m (finite) such that mth term onwards, Sn becomes and remains arbitrarily close to l, i.e. l is a limit point of the sequence. (ii) For any ε > 0, at the most a finite number of terms (depending on the choice of ε ) of the sequence can lie outside ] , [, l l − + ε ε i.e. there is at the most a finite number of n’s for which S l S l n n ≤ − ≥ + ε ε and . (iii) Since l S l n − < < + ε ε for all n m ≥ , therefore S l n < + ε , for infinite number of terms, i.e., infinite number of terms lie to the left of l + ε, or to the right of l − ε. It may therefore be observed that if we can find even one ε > 0 for which infinitely many terms of the sequence lie outside ] , [, l l − + ε ε then the sequence cannot converge to l. 3.2.4 Some Theorems Theorem 3.1. Every convergent sequence is bounded. Let a sequence {Sn} converge to the limit l. Let ε > 0 be a given number, so that ∃ a positive integer m such that S l n m n − < ∀ ≥ ε ⇔ − < < + ∀ ≥ l S l n m n ε ε . Let g l S S Sm = − − min , , , , . ε 1 2 1 ... n s
- 55. Discovering Diverse Content Through Random Scribd Documents
- 56. regarded as the chief means for attaining the spiritual end of the monastic life.” He calls his Rule “a very little rule for beginners”— minima inchoationis regula, and says that though there may be in it some things “a little severe,” still he hopes that he will establish “nothing harsh, nothing heavy.” The most cursory comparison between this new Rule and those which previously existed will make it abundantly clear that St. Benedict’s legislation was conceived in a spirit of moderation in regard to every detail of the monastic life. Common-sense, and the wise consideration of the superior in tempering any possible severity, according to the needs of times, places, and circumstances were, by his desire, to preside over the spiritual growth of those trained in his “school of divine service.” In addition to this St. Benedict broke with the past in another and not less important way, and in one which, if rightly considered and acted upon, more than compensated for the mitigation of corporal austerities introduced into his rule of life. The strong note of individualism characteristic of Egyptian monachism, which gave rise to what Dom Butler calls the “rivalry in ascetical achievement,” gave place in St. Benedict’s code to the common practices of the community, and to the entire submission of the individual will, even in matters of personal austerity and mortification, to the judgment of the superior. “This two-fold break with the past, in the elimination of austerity and in the sinking of the individual in the community, made St. Benedict’s Rule less a development than a revolution in monachism. It may be almost called a new creation; and it was destined to prove, as the subsequent history shows, peculiarly adapted to the new races that were peopling Western Europe.”[7] We are now in a position to turn to England. When, less than half a century after St. Benedict’s death, St. Augustine and his fellow monks in a.d. 597 first brought this Rule of Life to our country, a system of monasticism had been long established in the land. It was Celtic in its immediate origin; but whether it had been imported
- 57. originally from Egypt or the East generally, or whether, as some recent scholars have thought, it was a natural and spontaneous growth, is extremely doubtful. The method of life pursued by the Celtic monks and the austerities practised by them bear a singular resemblance to the main features of Egyptian monachism; so close, indeed, is this likeness that it is hard to believe there could have been no connection between them. One characteristic feature of Celtic monasticism, on the other hand, appears to be unique and to divide it off from every other type. The Celtic monasteries included among their officials one, and in some cases many bishops. At the head was the abbot, and the episcopal office was held by members of the house subordinate to him. In certain monasteries the number of bishops was so numerous as to suggest that they must have really occupied the position of priests at the subordinate churches. Thus St. Columba went in a.d. 590 from Iona to a synod at Drumcheatt, accompanied by as many as twenty bishops; and in some of the Irish ecclesiastical meetings the bishops, as in the case of some of the African synods, could be counted by hundreds. This Celtic system appears to be without parallel in other parts of the Christian Church, and scholars have suggested that it was a purely indigenous growth. One writer, Mr. Willis Bund, is of the opinion that the origin was tribal and that the first “monasteries” were mere settlements of Christians—clergy and laity, men, women, and children—who for the sake of protection lived together. It was at some subsequent date that a division was made between the male and female portions of the settlement, and later still the eremitical idea was grafted on the already existing system. If the tribal settlement was the origin of the Celtic monastery, it affords some explanation of the position occupied by the bishops as subjects of the abbots. The latter were in the first instance the chiefs or governors of the settlements, which would include the bishop or bishops of the churches comprised in the settlement. By degrees, according to the theory advanced, the head received a recognised ecclesiastical position as abbot, the bishop still continuing to occupy a subordinate position, although there is evidence in the lives of the
- 58. early Irish saints to show that the holder of the office was certainly treated with special dignity and honour. The Celtic monastic system was apparently in vogue among the remnant of the ancient British Church in Wales and the West Country on the coming of St. Augustine. Little is known with certainty, but as the British Church was Celtic in origin it may be presumed that the Celtic type of monachism prevailed amongst the Christians in this country after the Saxon conquest. Whether it followed the distinctive practice of Irish monasticism in regard to the position of the abbot and the subject bishops may perhaps be doubted, as this does not appear to have been the practice of the Celtic Church of Gaul, with which there was a close early connection. It has usually been supposed that the Rule of St. Columbanus represented the normal life of a Celtic monastery, but it has been lately shown that, so far as regards the Irish or Welsh houses, this Rule was never taken as a guide. It had its origin apparently in the fact that the Celtic monks on the Continent were induced, almost in spite of themselves, to adopt a mitigated rule of life by their close contact with Latin monasticism, which was then organising itself on the lines of the Rule of St. Benedict.[8] The Columban Rule was a code of great rigour, and “would, if carried out in its entirety, have made the Celtic monks almost, if not quite, the most austere of men.” Even if it was not actually in use, the Rule of St. Columbanus may safely be taken to indicate the tendencies of Celtic monasticism generally, and the impracticable nature of much of the legislation and the hard spirit which characterises it goes far to explain how it came to pass that whenever it was brought face to face with the wider, milder, and more flexible code of St. Benedict, invariably, sooner or later, it gave place to it. In some monasteries, for a time, the two Rules seem to have been combined, or at least to have existed side by side, as at Luxeuil and Bobbio, in Italy, in the seventh century; but when the abbot of the former monastery was called upon to defend the Celtic rule, at the Synod of Macon in a.d. 625,
- 59. the Columban code may be said to have ceased to exist anywhere as a separate rule of life. For the present purpose it will be sufficient to consider English monasticism from the coming of St. Augustine at the close of the sixth century as Benedictine. There was, it is true, a brief period when in Northumberland the Celtic form of regular observance established itself at Lindisfarne and elsewhere. This was due to the direct appeal made by King Edwy of Northumbria to the monks of Iona to come into Northumbria, and continue in the North the work of St. Paulinus, which had been interrupted by the incursions of Penda. Iona, the foundation and home of St. Columba, was a large monastic and missionary centre regulated according to the true type of Celtic monachism under the abbatial superior; and from Iona came St. Aidan and the other Celtic apostles of the northern parts. In one point, so far as the evidence exists for forming any judgment at all, the new foundation of Lindisfarne differed from the parent house at Iona. At the Northumbrian monastery the bishop was the head and took the place of the abbot, and did not occupy the subordinate position held by the bishops at Iona and its dependencies.
- 60. CHAPTER II THE MATERIAL PARTS OF A MONASTERY 1. THE CHURCH In any account of the parts of a monastic establishment the church obviously finds the first place. As St. Benedict laid down the principle that “nothing is to be preferred to the Opus Dei,” or Divine Service, so in every well-regulated religious establishment the church must of necessity be the very centre of the regular life as being, in fact no less than in word, the “House of God.” In northern climates the church was situated, as a rule, upon the northern side of the monastic buildings. With its high and massive walls it afforded to those who lived there a good shelter from the rough north winds. As the northern cloister usually stretched along the nave wall of the church and terminated at the south transept, the buildings of the choir and presbytery and also the retro-chapels, if there were any, gave some protection from the east wind. Sometimes, of course, there were exceptions, caused by the natural lie of the ground or other reason, which did not allow of the church being placed in the ordinary English position. Canterbury itself and Chester are examples of this, the church being in each case on the southern side, where also it is found very frequently in warm and sunny climates, with the obvious intention of obtaining from its high walls some shelter from the excessive heat of the sun. Convenience, therefore, and not any very recondite symbolism, may be considered to have usually dictated the position of “God’s house.”
- 61. Christian churches, especially the great cathedral and monastic churches, were originally designed and built upon lines which had much symbolism in them; the main body of the church with its transepts was to all, of course, a representation of Christ upon the cross. To the builders of these old sanctuaries the work was one of faith and love rather than a matter of mere mercenary business. They designed and worshipped whilst they wrought. To them, says one writer, the building “was instinct with speech, a tree of life planted in paradise; sending its roots deep down into the crypt; rising with stems in pillar and shaft; branching out into boughs over the vaulting; blossoming in diaper and mural flora; breaking out into foliage, flower, and fruit, on corbel, capital, and boss.” It was all real and true to them, for it sprang out of their strong belief that in the church they had “the House of God” and “the Gate of heaven,” into which at the moment of the solemn dedication “the King of Glory” had come to take lasting possession of His home. For this reason, to those who worshipped in any such sanctuary the idea that they stood in the “courts of the Lord” as His chosen ministers was ever present in their daily service, as with the eyes of their simple faith they could almost penetrate the veil that hid His majesty from their sight. As St. Benedict taught his disciples, mediæval monks believed “without any doubt” that God was present to them “in a special manner” when they “assisted at their divine service.” “Therefore,” says the great master of the regular observance, “let us consider in what manner and with what reverence it behoveth us to be in the sight of God and of the Angels, and so let us sing in choir, that mind and voice may accord together.”
- 62. Larger Image
- 63. NORWICH CATHEDRAL, WITH CLOISTERS So far as the religious life was concerned, the most important part of the church was of course the presbytery with the High Altar and the choir. Here all, or nearly all, public services were performed. The choir frequently, if not generally, stretched beyond the transepts and took up one, if not two, bays of the nave; being enclosed and divided off from that more public part by the great screen. Other gates of ironwork, across the aisle above the presbytery and in a line with the choir screen, kept the public from the south transept. Privacy was thus secured for the monks, whilst by this arrangement the people had full access to all parts of the sacred building except the choir and the transept nearest to the monastery.
- 64. The choir was entered, when the buildings were in the normal English position, from a door in the southern wall of the church at the juncture of the northern and eastern walks of the cloister. At the western end of the same northern cloister there was generally another door into the church reserved for the more solemn processions. The first, however, was the ordinary entrance used by the monks, and passing through it they found themselves in the area reserved for them within the screens which stretched across the choir and aisle. In the centre of the choir stood the great raised lectern or reading- desk, from which the lessons were chanted, and from which, also, the singing was directed by the cantor and his assistant. The stalls were arranged in two or more rows slightly raised one above the other. The superior and the second in command usually occupied the two stalls on each side of the main entrance furthest from the altar, the juniors being ranged nearest to the presbytery. This was the common practice except at the time of the celebration of the Sacrifice of the Holy Mass, or during such portion of the Office which preceded the Mass. On these occasions the elders took their places nearest to the altar, for the purpose of making the necessary oblations at the Holy Sacrifice. In many monastic choirs, for this reason, the abbot and prior had each two places reserved for their special use, one on either side near the altar, and the others at the entrance of the choir. Besides the great lectern of the choir there was likewise a second standing-desk for the reading of the Gospel at Matins, usually placed near to the steps of the presbytery. In some cases, apparently, this was always in its place, but more frequently it was brought into the choir for the occasion, and removed afterwards by the servers of the church.
- 65. Larger Image CANONS IN CHOIR There were in every church, besides the High Altar, several, and frequently numerous, smaller altars. The Rites of Durham describes minutely the nine altars arranged along the eastern wall of the church and facing the shrine of St. Cuthbert. “They,” says the author, “each had their several shrines and covers of wainscot over-head, in very decent and comely form, having likewise betwixt every altar a very fair and large partition of wainscot, all varnished over, with very fine branches and flowers and other imagery work
- 66. most finely and artificially pictured and gilded, containing the several lockers or ambers for the safe keeping of the vestments and ornaments belonging to every altar; with three or four aumbries in the wall pertaining to some of the said altars.” It would be now quite impossible to describe the rich adornments of an English mediæval monastic church. The Rites of Durham give some idea of the wealth of plate, vestments and hangings, and the art treasures, mural paintings and stained windows, with which generations of benefactors had enriched that great northern sanctuary. What we know of other monastic houses shows that Durham was not an exception in any way; but that almost any one, at any rate of the greater houses, could challenge comparison with it. A foreign traveller almost on the eve of their destruction speaks of the artistic wealth of the monastic churches of England as unrivalled by that of any other religious establishments in the whole of Europe. 2. THE CLOISTERS In every monastery next in public importance to the church came the cloisters. The very name has become a synonym for the monastery itself. The four walks of the cloister formed the dwelling- place of the community. With the progress of time there came into existence certain private rooms in which the officials transacted their business, and later still the use of private cells or cubicles became common, but these were the exception; and, at any rate, in England till the dissolution of the religious houses, the common life of the cloister was in full vigour.
- 67. THE CLOISTERS, WORCESTER In the normal position of the church on the north side of the monastic buildings, the north cloister with its openings looking south was the warmest of the four divisions. Here, in the first place, next the door of the church, was the prior’s seat, and the rest of the seniors in their order sat after him, not necessarily in order of seniority, but in the positions that best suited their work. The abbot’s place, “since his dignity demands,” as the Westminster Custumal puts it, was somewhat apart from the rest. He had his fixed seat at the end of the eastern cloister nearest to the church door. In the same cloister, but more towards the other, or southern end, the novice-master taught his novices, and the walk immediately opposite, namely, the western side of the cloister, was devoted to the junior monks, who were, as the Rule of St. Benedict says, “adhuc in custodia”: still under stricter discipline. The southern walk, which would have been in ordinary circumstances the sunless, cold
- 68. side of the quadrangle, was not usually occupied in the daily life of the community. This was the common position for the refectory, with the lavatory close at hand, and the aumbries or cupboards for the towels, etc. It was here also that the door from the outside world into the monastic precincts was usually to be found. At Durham, for example, we are told that— “there was on the south side of the cloister door, a stool, or seat with four feet, and a back of wood joined to the said stool, which was made fast in the wall for the porter to sit on, which did keep the cloister door. And before the said stool it was boarded in under foot, for warmness. And he that was the last porter there was called Edward Pattinson.” The same account describes the cupboards near to the refectory door in which the monks kept their towels— “All the forepart of the aumbry was thorough carved work, to give air to the towels.” There were “three doors in the forepart of either aumbry and a lock on every door, and every monk had a key for the said aumbries, wherein did hang in every one clean towels for the monks to dry their hands on, when they washed and went to dinner.” We who see the cold damp-stained cloisters of the old monastic buildings as they are to-day, as at Westminster for example, may well feel a difficulty in realising what they were in the time of their glory. Day after day for centuries the cloister was the centre of the activity of the religious establishment. The quadrangle was the place where the monks lived and studied and wrote. In the three sides— the northern, eastern, and western walks—were transacted the chief business of the house, other than what was merely external. Here the older monks laboured at the tasks appointed them by obedience, or discussed questions relating to ecclesiastical learning or regular observance, or at permitted times joined in recreative conversation. Here, too, in the parts set aside for the purpose, the younger
- 69. members toiled at their studies under the eye of their teacher, learnt the monastic observance from the lips of the novice-master, or practised the chants and melodies of the Divine Office with the cantor or his assistant. How the work was done in the winter time, even supposing that the great windows looking out on to the cloister-garth were glazed or closed with wooden shutters, must ever remain a mystery. In some places, it is true, certain screenwork divisions appear to have been devised, so as to afford some shelter and protection to the elder members and scribes of the monastery from the sharper draughts inevitable in an open cloister. The account given in the Rites of Durham on this point is worth quoting at length:— “In the cloister,” says the writer—and he is speaking of the northern walk, set apart for the seniors—“in the cloister there were carrels finely wainscotted and very close, all but the forepart, which had carved work to give light in at their carrel doors. And in every carrel was a desk to lie their books on, and the carrel was no greater than from one stanchell (centre-bar) of the window to another. And over against the carrels, against the church wall, did stand certain great aumbries of wainscot all full of books, with great store of ancient manuscripts to help them in their study.” In these cupboards, “did lie as well the old ancient written Doctors of the Church as other profane authors, with divers other holy men’s works, so that every one did study what doctor pleased him best, having the Library at all times to go and study in besides these carrels.”
- 70. THE CLOISTERS, GLOUCESTER, SHEWING CARRELS In speaking of the novices the same writer tells us that— “over against the said treasury door was a fair seat of wainscot, where the novices were taught. And the master of the novices had a pretty seat of wainscot adjoining to the south side of the treasury door, over against the seat where the novices sat; and there he taught the novices both forenoon and afternoon. No strangers or other persons were suffered to molest, or trouble the said novices, or monks in their carrels while they were at their
- 71. books within the cloister. For to this purpose there was a porter appointed to keep the cloister door.” In other monasteries, such for example as Westminster and St. Augustine’s, Canterbury, these enclosed wooden sitting-places seem to have been very few in number, and allowed only to those officers of the house who had much business to transact for the common good. At Durham, however, we are told that “every one of the old monks” had his own special seat, and in each window of the south cloister there were set “three of these pews or carrels.” 3. THE REFECTORY The refectory, sometimes called the fratry or frater-house, was the common hall for all conventual meals. Its situation in the plan of a monastic establishment was almost always as far removed from the church as possible, that is, it was on the opposite side of the cloister quadrangle and, according to the usual plan, in the southern walk of the cloister. The reason for this arrangement is obvious. It was to secure that the church and its precincts might be kept as free as possible from the annoyance caused by the noise and smells necessarily connected with the preparation and consumption of the meals. As a rule, the walls of the hall would no doubt have been wainscotted. At one end, probably, great presses would have been placed to receive the plate and linen, with the salt-cellars, cups, and other ordinary requirements for the common meals. The floor of a monastic refectory was spread with hay or rushes, which covering was changed three or four times in the year; and the tables were ranged in single rows lengthways, with the benches for the monks upon the inside, where they sat with their backs to the panelled walls. At the east end, under some sacred figure, or painting of the crucifix, or of our Lord in glory, called the Majestas, was the mensa
- 72. major, or high table for the superior. Above this the scylla or small signal-bell was suspended. This was sounded by the president of the meal as a sign that the community might begin their refection, and for the commencement of each of the new courses. The pulpit, or reading-desk, was, as a rule, placed upon the south side of the hall, and below it was usually placed the table for the novices, presided over by their master. “At which time (of meals),” says the Rites of Durham, “the master observed this wholesome order for the continual instructing of their youth in virtue and learning; that is, one of the novices, at the election and appointment of the master, did read some part of the Old and New Testament, in Latin, in dinner-time, having a convenient place at the south end of the high table within a fair glass window, environed with iron, and certain steps of stone with iron rails of the one side to go up into it and to support an iron desk there placed, upon which lay the Holy Bible.” In most cases the kitchens and offices would have been situated near the western end of the refectory, across which a screen pierced with doors would probably have somewhat veiled the serving-hatch, the dresser, and the passages to the butteries, cellars, and pantry.
- 73. THE REFECTORY, CLEVE ABBEY Besides the great refectory there was frequently a smaller hall, called by various names such as the “misericord,” or “oriel” at St. Alban’s, the “disport” (deportus) at Canterbury, and the “spane” at Peterborough. In this smaller dining-place those who had been bled and others, who by the dispensation of the superior were to have different or better food than that served in the common refectory, came to their meals. At Durham, apparently, the ordinary dining- place was called the “loft,” and was at the west end of a larger hall entered from the south alley of the cloister, called the “frater-house.” In this hall “the great feast of Saint Cuthbert’s day in Lent was holden.” In an aumbry in the wainscot, on the left-hand of the door, says the author of the Rites of Durham, was kept the great mazer, called the grace-cup, “which did service to the monks everyday, after grace was said, to drink in round the table.” 4. THE KITCHEN
- 74. Near to the refectory was, of course, the conventual kitchen. At Canterbury this kitchen was a square of some forty-five feet; at Durham it was somewhat smaller; and at Glastonbury, Worcester, and Chester the hall was some thirty-five feet square. A small courtyard with the usual offices adjoined it; and this sometimes, as at Westminster and Chester, had a tower and a larder on the western side. According to the Cluniac constitutions there were to be two kitchens: the one served in weekly turns by the brethren, the other in which a good deal of the food was prepared by paid servants. The first was chiefly used for the preparation of the soup or pottage, which formed the foundation of the monastic dinner. The furniture of this kitchen is minutely described in the Custumals: there were to be three caldaria or cauldrons for boiling water: one for cooking the beans, a second for the vegetables, and a third, with an iron tripod to stand it upon, to furnish hot water for washing plates, dishes, cloths, etc. Secondly, there were to be four great dishes or vessels: one for half-cooked beans; another and much larger one, into which water was always to be kept running, for washing vegetables; a third for washing up plates and dishes; and a fourth to be reserved for holding a supply of hot water required for the weekly feet-washing, and for the shaving of faces and tonsures, etc. In the same way there were to be always in the kitchen four spoons: the first for beans, the second for vegetables, the third (a small one naturally) for seasoning the soup, and the fourth (an iron one of large size) for shovelling coals on to the fire. Besides these necessary articles, the superior was to see that there were to be always at hand four pairs of sleeves for the use of the servers, that they might not soil their ordinary habits; two pairs of gloves for moving hot vessels, and three napkins for wiping dishes, etc., which were to be changed every Thursday. Besides these things there were, of course, to be knives, and a stone wherewith to sharpen them; a small dish to get hot water quickly when required; a strainer; an urn to draw hot water from; two ladles; a fan to blow the fire up when needed, and stands to set the pots upon, etc.
- 75. Larger Image The work of the weekly cooks is also carefully set out in these constitutions. These officials were four in number, and, upon the sign for vespers, after making their prayer, they were to proceed to the kitchen and obtain the necessary measure of beans for the following day. They then said their vespers together, and proceeded to wash the beans in three waters, putting them afterwards into the great boiling-pot with water ready for the next day. After Lauds on the following day, when they had received the usual blessing for the servers, after washing themselves they proceeded to the kitchen and set the cauldron of beans on the fire. The pot was to be watched most carefully lest the contents should be burnt. The skins were to be taken off as they became loosened, and the beans were to be removed as they were cooked. When all had been finished, the great cauldron was to be scoured and cleaned “usque ad nitidum.” Directly
- 76. the beans had been removed from the fire, another pot was to be put in its place, so that there might always be a good supply of water for washing plates and dishes. These, when cleaned, were to be put into a rack to dry; this rack was to be constantly and thoroughly scoured and kept clean and sweet. When the cooking of this bean soup had progressed so far, the four cooks were to sit down and say their Divine Office together whilst the hot water was being boiled. A third pot, with vegetables in cold water, was to be then made ready to take its place on the fire, after the Gospel of the morning Mass. When the daily Chapter, at which all had to be present, was finished, the beans were again to be put on the fire and boiled with more water, whilst the vegetables also were set to cook; and when these were done the cooks got the lard and seasoning, and, having melted it, poured it over them. Two of the four weekly cooks now went to the High Mass, the other two remaining behind to watch the dinner and to put more water into the cooking-pots when needed. When the community were ready for their meal, the first cook ladled out the soup into dishes, and the other three carried them to the refectory. In the same way the vegetables were to be served to the community, and when this had been done the four weekly cooks proceeded at once to wash with hot water the dishes and plates which had been used for beans and vegetables, lest by delay any remains should stick to the substance of the plate and be afterwards difficult to remove. 5. THE CHAPTER-HOUSE The chapter-hall, or house, was situated on the eastern side of the cloister, as near to the church as possible. Its shape, usually rectangular, sometimes varied according to circumstances and places. At Worcester and Westminster, for example, it was octagonal; at Canterbury and Chester rectangular; at Durham and Norwich rectangular with an apsidal termination. Seats were
- 77. arranged along the walls for the monks, sometimes in two rows, one raised above the other, and at the easternmost part of the hall was the chair of the superior, with the crucifix or Majestas over it. In the centre a raised desk or pulpit was arranged for the reader of the Martyrology, etc., at that part of Prime which preceded the daily Chapter, and at the evening Collation before Compline. THE CHAPTER HOUSE, WESTMINSTER 6. THE DORMITORY
- 78. The position of the dormitory among the claustral buildings was apparently not so determined either by rule or custom, as some of the other parts of the religious house. Normally, it may be taken to have communicated with the southern transept, for the purpose of giving easy access to the choir for the night offices. In two cases it stood at right angles to the cloister—at Worcester on the western side, and at Winchester on the east. The Rites of Durham says that “on the west side of the cloister was a large house called the Dortor, where the monks and novices lay. Every monk had a little chamber to himself. Each chamber had a window towards the Chapter, and the partition betwixt every chamber was close wainscotted, and in each window was a desk to support their books.” The place itself at Durham, and, indeed, no doubt, usually, was raised upon an undercroft and divided into various chambers and rooms. Amongst these were the treasury at Durham and Westminster, and the passage to the chapter-hall in the latter. The dormitory-hall was originally one open apartment, in which the beds of the monks were placed without screens or dividing hangings. In process of time, however, divisions became introduced such as are described by the author of the Rites of Durham, and such as we know existed elsewhere. The cubicles or cells thus formed came to be used for the purpose of study as well as for sleeping, which accounts for the presence of the “desk to support their books” spoken of above. The dormitory also communicated with the latrine or rere-dortor, which was lighted, partitioned, and provided with clean hay. For the purpose of easy access, as for instance at Worcester, the dormitory frequently communicated directly with the church through the south-western turret; at Canterbury a gallery was formed in the west gable-wall of the chapter-house, over the doorway, and continuing over the cloister roof, came out into an upper chapel in the northern part of the transept; at Westminster a bridge crossed the west end of the sacristy, and at St. Alban’s and Winchester
- 79. passages in the wall of the transept gave communication by stairs into the church. 7. THE INFIRMARY In the disposition of the parts of the religious house no fixed locality was apparently assigned by rule or custom to the infirmary, or house for the sick and aged. Usually it appears to have been to the east of the dormitory; but there were undoubtedly numerous exceptions. At Worcester it faced the west front of the church, and at Durham and Rochester apparently it joined it; whilst at Norwich and Gloucester it was in a position parallel to the refectory. Adjoining the infirmary was sometimes the herbarium, or garden for herbs; and occasionally, as at Westminster, Gloucester, and Canterbury, this was surrounded by little cloisters. The main hall, or large room, of the infirmary often included a chapel at the easternmost point, where the sick could say their Hours and other Offices when able to do so, and where the infirmarian could say Mass for those under his charge. According to the constitutions of all religious bodies the care of the sick was enjoined upon the superior of every religious house as one of his most important duties. “Before all things, and above all things,” says St. Benedict in his Rule, “special care must be taken of the sick, so that they be served in very deed, as Christ Himself, for He saith: ‘I was sick, and ye visited me’; and, ‘What ye did to one of these My least Brethren, ye did to Me.’” On this principle not only was a special official appointed in every monastery, whose first duty it was to look to the care and comfort of those who were infirm and sick, but the officials of the house generally were charged with seeing that they were supplied with what was needed for their comfort and cure. Above all, says the great legislator, “let the abbot take special care they be not
- 80. neglected,” that they have what they require at the hands of the cellarer, and that the attendants do not neglect them, “because,” he adds, “whatever is done amiss by his disciples is imputed to him.” For this reason, at stated times, as for instance immediately after the midday meal, the superior, who had presided in the common refectory, was charged to visit the sick brethren in the infirmary, in order to be sure that they had been served properly and in no ways neglected. 8. THE GUEST-HOUSE The guest-house (hostellary, hostry, etc.) was a necessary part of every great religious house. It was presided over by a senior monk, whose duty it was to keep the hall and chambers ready for the reception of guests, and to be ever prepared to receive those who came to ask for hospitality. Naturally the guest-house was situated where it would be least likely to interfere with the privacy of the monastery. The guest-place at Canterbury was of great size, measuring forty feet broad by a hundred and fifty feet long. The main building was a big hall, resembling a church with columns, having on each side bedrooms or cubicles leading out of it. In the thirteenth century John de Hertford, abbot of St. Alban’s, built a noble hall for the use of guests frequenting his abbey, with an inner parlour having a fireplace in it, and many chambers arranged for the use of various kinds of guests. It had also a pro-aula, or reception- room, in which the guest-master first received the pilgrim or traveller, before conducting him to the church, or arranging for a reception corresponding to his rank and position. In the greater monastic establishments there were frequently several places for the reception of guests. The abbot, or superior, had rooms to accommodate distinguished or honoured guests and benefactors of the establishment. The cellarer’s department, too, frequently had to entertain merchants and others who came upon business of the
- 81. house: a third shelter was provided near the gate of the monastery for the poorer folk, and a fourth for the monks of other religious houses, who had their meals in the common refectory, and joined in many of the exercises of the community. The Rites of Durham thus describes the guest-house which the author remembered in the great cathedral monastery of the North:— “There was a famous house of hospitality, called the Guest Hall, within the Abbey garth of Durham, on the west side, towards the water, the Terrar of the house being master thereof, as one appointed to give entertainment to all states, both noble, gentle, and whatsoever degree that came thither as strangers, their entertainment not being inferior to any place in England, both for the goodness of their diet, the sweet and dainty furniture of their lodgings, and generally all things necessary for travellers. And, withal, this entertainment continuing, (the monks) not willing or commanding any man to depart, upon his honest and good behaviour. This hall is a goodly, brave place, much like unto the body of a church, with very fair pillars supporting it on either side, and in the midst of the hall a most large range for the fire. The chambers and lodgings belonging to it were sweetly kept and so richly furnished that they were not unpleasant to lie in, especially one chamber called the ‘king’s chamber,’ deserving that name, in that the king himself might very well have lain in it, for the princely linen thereof.... The prior (whose hospitality was such as that there needed no guest-hall, but that they (the Convent) were desirous to abound in all liberal and free almsgiving) did keep a most honourable house and very noble entertainment, being attended upon both with gentlemen and yeomen, of the best in the country, as the honourable service of his house deserved no less. The benevolence thereof, with the relief and alms of the whole Convent, was always open and
- 82. free, not only to the poor of the city of Durham, but to all the poor people of the country besides.” In most monastic statutes, the time during which a visitor was to be allowed free hospitality was not unlimited, as, according to the recollection of the author of the Rites of Durham, appears to have been the case in that monastery. The usual period was apparently two days and nights, and in ordinary cases after dinner on the third day the guest was expected to take his departure. If for any reason a visitor desired to prolong his stay, permission had to be obtained from the superior by the guest-master. Unless prevented by sickness, after that time the guest had to rise for Matins, and otherwise follow the exercises of the community. With the Franciscans, a visitor who asked for hospitality from the convent beyond three days, had to beg pardon in the conventual chapter before he departed for his excessive demand upon the hospitality of the house. 9. THE PARLOUR OR LOCUTORIUM In most Custumals of monastic observance mention is made of a Parlour, and in some of more than one such place. Here the monks could be sent for by the superiors to discuss necessary matters of business, when strict silence had to be observed in the cloister itself. Here, too—it may be in the same, or in another such room—visitors could converse with the religious they had come to see. Sometimes, apparently, among the Cistercians, the place where the monastic schools were held, other than the cloister, was called the auditorium or locutorium. At Durham, the room called the parlour stood between the chapter-house and the church door, and is described as “a place for merchants to utter their wares.” It apparently had a door which gave access to the monastic cemetery, as the religious were directed to pass through it for the funeral of any of the brethren. During the times of silence, when anything had to be settled without unnecessary delay, the officials could summon any of the religious to
- 83. the parlour for the purpose; but they were warned not to make any long stay, and to take great care that no sound of their voices disturbed the quiet of the cloister. 10. THE ALMONRY No religious house was complete without a place where the poor could come and beg alms in the name of Christ. The convent doles of food and clothing were administered by one of the senior monks, who, by his office of almoner, had to interview the crowds of poor who daily flocked to the gate in search of relief. His charity was to be wider than his means; and where he could not satisfy the actual needs of all, he was at least to manifest his Christian sympathy for their sufferings. The house or room, from which the monastic relief was given, frequently stood near the church, as showing the necessary connection between charity and religion. In most of the almonries, at any rate in those of the larger monasteries, there was a free school for poor boys. It was in these that most of the students who were presented for Ordination by the religious houses in such number during the fourteenth and fifteenth centuries, (as is shown by the episcopal registers of the English dioceses), were prepared to exercise their sacred ministry in the ranks of the parochial clergy. 11. THE COMMON-ROOM OR CALEFACTORY The common-room, sometimes called the calefactory or warming- place, was a room to which the religious resorted, especially in winter, for the purpose of warming themselves at the common fire, which was lighted on the feast of All Saints, November 1st, and kept burning daily until Easter. On certain occasions, such as Christmas night, when the Offices in the church were specially long, the
- 84. caretaker was warned to be particularly careful to have a bright fire burning for the community to go to when they came out of the choir. The common-room was also used at times for the purpose of recreation. “On the right hand, as you go out of the cloisters into the infirmary,” says the Rites of Durham, “was the Common House and a master thereof. This house was intended to this end, to have a fire kept in it all the winter, for the monks to come and warm them at, being allowed no fire but that only, except the masters and officers of the house, who had their several fires. There was belonging to the Common House a garden and a bowling alley, on the back-side of the said house, towards the water, for the novices sometimes to recreate themselves, when they had leave of their master; he standing by to see their good order. “Also, within this house did the master thereof keep his O Sapientia once a year—namely, between Martinmas and Christmas—a solemn banquet that the prior and convent did use at that time of the year only, when their banquet was of figs and raisins, ale and cakes; and thereof no superfluity or excess, but a scholastical and moderate congratulation amongst themselves.” 12. THE LIBRARY “A monastery without a library is like a castle without an armoury” was an old monastic saying. At first, and in most places in England probably to the end, there was no special hall, room, or place which was set aside for the reception of the books belonging to the monastery. In the church and in the cloister there were generally cupboards to hold the manuscripts in constant use. It was not till the
- 85. later middle ages that the practice of gathering together the books of an establishment into one place or room became at all common. At Durham, about 1446, Prior Wessington made a library, “well replenished with old written doctors and other Histories and Ecclesiastical writers,” to which henceforth the monks could always repair to study in, “besides their carrels” in the cloister. So, too, at St. Alban’s, Michael de Mentmore, who was abbot from 1335 to 1349, besides enriching the presses in the cloister with books, made a collection of special volumes in what he called his study. This collection grew; but it was not till 1452 that Abbot Whethamstede finally completed the library, which had long been projected. About the same time, at Canterbury, Prior Thomas Goldstone finished a library there, which was enriched by the celebrated Prior William Sellyng with many precious classical manuscripts brought back from Italy. In the same way many other religious houses in the fifteenth century erected, or set apart, special places for their collections of books, whilst still retaining the great cloister presses for those volumes which were in daily and constant use. Larger Image
- 86. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com