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 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.