On a New Proof of the Key Step in the Proof of Brouwer’s Fixed Point Theorem

We present a solution of Exercise 1.2.1 in [1] 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 this note 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.