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

## TL;DR

This paper provides a straightforward, elementary proof demonstrating that the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma implies Sperner's lemma, clarifying a previously less direct logical connection among key economic and topological results.

## Contribution

It offers the first elementary direct proof that the KKM lemma implies Sperner's lemma, filling a gap in the literature of foundational mathematical results.

## Key findings

- Established a direct implication from KKM lemma to Sperner's lemma
- Simplified the understanding of the relationship among key topological lemmas
- Enhanced the toolkit for proofs in economic theory and topology

## Abstract

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