, Let n tend to infinity, there is a subsequence of (x n ) converging to a point x * which is actually a fixed-point of ?. Florenzano (1981), in Proposition 2, also makes use the Brouwer fixed point theorem to prove the Kakutani fixed point theorem. More precisely, for any > 0, Florenzano considers a covering of ? by a finite family of open balls and defines the function f as in our above proof. By applying the Brouwer fixed point theorem, f has a fixed point x . Let ? 0, then x ?x. To prove thatx ? ?(x), assume that this is not a case, then apply the Separation Theorem to the sets {x} and ?(x) to get a contradiction. We proceed as in Florenzano (1981) but use the Sperner lemma to get a fixed point x of the function f . Let ? 0, then x ?x, 1941.
, Remark 7 (The Kakutani fixed point theorem and the Gale-Nikaido-Debreu lemma)
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