Compact subset of a polytope is contained in a finite subcomplex.

Let A be a compact subset of |K|.
Suppose it is not contained in any finite subcomplex of K.
First note that interiors of simplices in K are disjoint. A can't be covered by finite number of interiors of simplices in K, so if we pick an element x_s whenever an interior of a simplex s intersects A, then the set B of all such x_s is infinite. Furthermore, if B^' ⊆ B is any subset, then any simplex in K intersects B^' only at finitely many points, so it is closed in |K|, and hence in B. i.e., B is a discrete space. Also B itself is closed in A, so is compact. i.e., we have an infinite compact discrete space B. This can't be true because an infinite compact space must have a limit point. QED