Prove the K-K-M theorem directly, using Sperner's lemma. Let A0, A1,..., An be closed subsets of an

Question:

Prove the K-K-M theorem directly, using Sperner's lemma.
Let A0, A1,..., An be closed subsets of an n-dimensional simplex S with vertices x0, x1,..., xn. If for every I ⊆ {0, 1,..., n} the face conv{xi: i ∊ I} is contained in the corresponding union ⋃i∊I Ai, , then the intersection ∩ni=0 Ai is nonempty.