On the (in)equivalence of Brouwer's fixed point theorem and Sperner's lemma

Abstract

We consider Brouwer's fixed point theorem and Sperner's lemma in one dimension. We present a proof of the Brouwer theorem using the Sperner lemma, and vice versa. However, we also show that they are not equivalent, because the Sperner lemma holds in the ordered field of rational numbers, whereas proving the Brouwer theorem requires the property of completeness.