Every compact subset can intersect only finitely many simplices

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Well my question is almost in the title: how can I prove that every compact subset C of |K| can intersect only finitely many simplices, where K is a infinite simplicial complex?

In particular given C I need to find a subcomplex L of K finite s.t. |L| contains C.

I’m not an expert of homology and I need all the proof, because I saw some hints in other questions, but it wasn’t enought. Thanks a lot.