the proof.

Topological proof of the base case

Let , and consider the cover. Since the intersection of convex sets is convex, every non-empty finite intersection of elements of is contractible. Hence is a good cover.

By the basic nerve lemma, is homotopy equivalent to the geometric realization of the nerve of the cover. Assuming that

the nerve is the boundary complex of the simplex consisting of all the -faces. Therefore is

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