# Constructive proof of the approximate Brouwer's Fixed Point Theorem for $\Delta^n$

The approximate Brouwer Fixed Point Theorem (aBFPT) for the standard $$n$$-simplex is:

Let $$f$$ be a uniformly continuous function from $$\Delta^n$$ into itself. Then for each $$\varepsilon>0$$ there exists $$x\in\Delta^n$$ such that $$d(f(x),x)<\varepsilon$$.

Does anyone know a reference for a full proof of aBFPT via Sperner's Lemma in Bishop-style constructive mathematics (BISH)?

A constructive proof of aBFPT via Gale's Hex Theorem is in Hendtlass's PhD thesis. A constructive proof for the $$2$$-simplex is in this paper by van Dalen. That's all I could find.

If you don't know a reference but have a proof handy I'd very much like to see that as well.