# A proof that Sperner's Lemma implies KKM?

Wikepedia KKM says that Brouwer fixed-point theorem, Sperner's lemma, and Knaster–Kuratowski–Mazurkiewicz lemma are equivalent.

In "A course of topological combinatorics", I find a proof that Sperner implies Brouwer. I am wondering is there a proof that Sperner implies KKM?

Consider the $$k$$th barycentric division, which is a triangulation, $$T^k$$ of $$\Delta_n:=\{(x_1,...,x_{n+1}): x_i\ge 0, \sum x_i=1\}$$.
We color each vertex $$v$$ of $$T_k$$ by minimum $$i$$ such that $$v_i\neq 0$$ and $$v\in A_i$$, where $$A_1,...,A_n$$ are the closed sets in KKM lemma. Then this coloring is a legal Sperner coloring and we know that there exists a simplex $$\sigma_k$$ in $$T^k$$ whose vertices are rainbow (in $$n$$ colors).
As the volume of $$\sigma_k$$ tends to 0 as $$k$$ goes to infinity, we know that there exists $$z\in \Delta_n$$ that is a common accumulation point of all the vertices in an infinite subsequence of $$\sigma_k$$ (say $$\sigma_{k_\ell}$$). As $$A_i$$ are closed and the vertices of $$\sigma_k$$ are rainbow, we know that $$z\in\bigcap_i A_i$$. (For the vertices $$v(k_\ell,i)$$, which are the vertices in $$\sigma_{k_\ell}$$ and are colored by $$i$$, by definition we have $$v(k_\ell,i)\in A_i$$, and we have $$v=\lim_\ell v(k_\ell,i)$$ is also in $$A_i$$ as $$A_i$$ are closed.)