On a new proof of the key step in the proof of Brouwer's fixed point theorem

TL;DR

This paper introduces a novel, simple proof of a crucial step in Brouwer's fixed point theorem, avoiding complex algebraic or differential topology methods, and potentially broadening understanding of the theorem.

Contribution

The paper provides a new, elementary proof of a key step in Brouwer's fixed point theorem, differing from traditional algebraic or differential approaches.

Findings

New proof simplifies understanding of the key step

Approach is based on a simple, overlooked observation

Potentially broadens accessibility of Brouwer's fixed point theorem

Abstract

We present a solution of Exercise 1.2.1 of [2] which yields a short new proof of a key step in one of proofs of Brouwer's fixed point theorem, 1910. A few people asked the author about the details of the solution and they might be interesting to a broader audience. Our approach is absolutely different from the ones using algebraic or differential topology or differential calculus and is based on a simple observation which somehow escaped many authors treating this theorem in the past.