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Three central results in economic theory --- Brouwer's fixed-point theorem,Sperner's lemma, and the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma --- areknown to be equivalent. In almost all cases, elementary direct proofs of one ofthese results using any of the others are easily found in the literature. Thisseems not to be the case for the claim that the KKM lemma implies Sperner'slemma. The goal of this note is to provide such a proof.

An elementary direct proof that the Knaster-Kuratowski-Mazurkiewicz lemma implies Sperner's lemma