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- Then, since Xxn pn (`), there exists a binary lottery Xxn pn such that ` < xn < xn, 39 Documents de travail du Centre d'Economie de la Sorbonne - 2014.54RR (Version révisée) and Xxn pn (`). Let xn+1 and pn+1 be de…ned by X xn+1 pn+1 Xxn pn . Since fX xn pn gn2N converges towards (`), there exists an integer N, such that m N ) ` xm < xn+1 and pm pn+1. This implies that X xn+1 pn+1 should be preferred to the X xm pm s and, consequently, that (`) should be preferred to the X xm pm s, what contradicts the fact that fX xn pn gn2N is decreasing and converges towards (`). Hence ` = a and fSngn2N converges towards = u(w). As a consequence, equality (31) is checked.
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