Constructive Proof of Brouwer’s Fixed Point Theorem for Sequentially Locally Non-constant and Uniformly Sequentially Continuous Functions

We present a constructive proof of Brouwer’s fixed point theorem for sequentially locally non-constant and uniformly sequentially continuous functions based on the existence of approximate fixed points. And we will show that our Brouwer’s fixed point theorem implies Sperner’s lemma for a simplex. Since the existence of approximate fixed points is derived from Sperner’s lemma, our Brouwer’s fixed point theorem is equivalent to Sperner’s lemma.