, 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)

C. D. Aliprantis and K. C. Border, Infinite dimensional analysis: a Hitchhiker's guide, third Edition, 2006.

K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, vol.22, pp.265-290, 1954.

H. Ben-el-mechaiekh, F. Bich, and M. Florenzano, General equilibrium and fixed point theory : A partial survey, CES working paper series, 2009.

C. Berge, Espaces topologiques et fonctions multivoques, 1959.

E. Bishop and D. Bridges, Constructive Analysis, 1985.

D. Bridges and L. Vita, Techniques of Constructive Mathematics, 2006.

L. E. Brouwer, Über Abbildung von Mannigfaltigkeiten, Mathematische Annalen 71, pp.97-115, 1911.

K. C. Border, Fixed point theorems with applications to economics and game theory, 1985.

C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, vol.64, pp.95-115, 1907.

D. Cass, Competitive equilibrium with incomplete financial markets, Journal of Mathematical Economics, vol.42, pp.384-405, 2006.

A. Cellina, A theorem on the approximation of compact multivalued mappings, Atti. Mat. Naz. Lincei, vol.8, pp.149-153, 1969.

D. I. Cohen, On Sperner lemma, Journal of Combinatorial Theory, vol.2, pp.585-587, 1967.

G. Debreu, A social equilibrium existence theorem, Proceedings of the National Academy of Sciences, vol.38, issue.10, pp.886-893, 1952.

G. Debreu, Market equilibrium, Proceedings of the National Academy of Sciences, vol.42, issue.11, pp.876-878, 1956.

G. Debreu, Theory of Value -An Axiomatic Analysis of Economic Equilibrium, 1959.

G. Debreu, Existence of competitive equilibrium, In Handbook of Mathematical Economics, vol.II, p.15, 1982.

D. Gale and A. Mas-colell, An equilibrium existence theorem for a general model without ordered preferences, Journal of Mathematical Economics, vol.2, pp.9-15, 1975.

D. Gale and A. Mas-colell, Corrections to an equilibrium existence theorem for a general model without ordered preferences, Journal of Mathematical Economics, vol.6, pp.297-298, 1975.

M. Florenzano, L'Équilibreéconomique général transitif et intransitif: problemes d'existence, 1981.

M. Florenzano, The Gale-Nikaido-Debreu lemma and the existence of transitive equilibrium with or without the free-disposal assumption, Journal of Mathematical Economics, vol.9, pp.113-134, 1982.

M. Florenzano, General equilibrium of financial markets: An introduction. CES working paper series, 1999.

M. Florenzano, General Equilibrium Analysis: Existence and Optimality Properties of Equilibria, 2003.

M. Florenzano, Two lemmas that changed general equilibrium theory. CES working paper series, 2009.
URL : https://hal.archives-ouvertes.fr/halshs-00423878

M. Florenzano and C. Van, A note on the Gale-Nikaido-Debreu Lemma and the Existence of General Equilibrium, vol.22, pp.107-110, 1986.

M. Florenzano, L. Van, and C. , Finite Dimensional Convexity and Optimization, 2001.

D. Gale, The law of supply and demand, Mathematica Scandinavica, vol.3, pp.155-169, 1955.

J. Hadamard, Note sur quelques applications de l'indice de Kronecker, Introductio? a la théorie des fonctions d'une variable, pp.437-477, 1910.

S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Mathematical Journal, vol.8, issue.3, pp.457-459, 1941.

B. Knaster, K. Kuratowski, and S. Mazurkiewicz, A Ein Beweis des Fixpunktsatzes für n-Dimensionale Simplexe, Fund. Math, vol.14, pp.132-137, 1929.

H. W. Kuhn, Simplicial aproximations of fixed points, Proc. Nat. Acad: Sci. U.S.A, vol.61, pp.1238-1242, 1968.

L. Van and C. , Topological degree and the Sperner lemma, Journal of Optimization Theory and Applications, vol.37, pp.371-377, 1982.

P. Mackowiak, The existence of equilibrium in a simple exchange model, Fixed Point Theory and Applications 104, 2013.

M. Magill and M. Quinzii, Theory of Incomplete Markets, vol.1, 1996.

L. W. Mckenzie, On the existence of general equilibrium for a competitive market, Econometrica, vol.27, pp.54-71, 1959.

E. Michael, Continuous selections. I, vol.63, pp.361-382, 1956.

H. Nikaido, On the classical multilateral exchange problem, Metroeconomica, vol.8, pp.135-145, 1956.

S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam Journal of Mathematics, vol.27, pp.187-222, 1999.

S. Park and K. S. Jeong, The proof of the Sperner lemma from the Brouwer fixed point theorem, 2003.

H. Scarf, The Approximation of Fixed Points of a Continuous Mapping, SIAM Journal on Applied Mathematics, vol.15, pp.1328-1343, 1967.

H. Scarf and T. Hansen, The Computation of Economic Equilibria, 1973.

H. Scarf, The Computation of Equilibrium Prices: An Exposition, In Handbook of Mathematical Economics, vol.II, 1982.

E. Sperner, Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Seminar Univ. Hambourg, vol.6, pp.265-272, 1928.

Y. Shmalo, Combinatorial Proof of Kakutani's Fixed Point Theorem, 2018.

M. Sondjaja, Sperner lemma Implies Kakutani's Fixed Point Theorem, HMC Senior Theses, vol.214, 2008.

F. E. Su, Rental Harmony: Sperner lemma in Fair Division, The American Mathematical Monthly, vol.106, issue.10, pp.930-942, 1999.

H. Uzawa, Walras existence theorem and Brouwer's fixed point theorem, Economic Studies Quarterly, vol.13, pp.59-62, 1962.

J. Von-neumann, Über ein Okonomisches Gleichungs-System und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes, Ergebnisse eines Mathematischen Kolloquiums, pp.73-83, 1937.

L. Walras, Eléments d'économie politique pure, pp.1874-1877, 1877.

M. Yoseloff, Topological proofs of some combinatorial theorems, Journal of Combinatorial Theory Series A, vol.17, pp.95-111, 1974.