How to prove convex hull of compact set is compact

Question

seems straightforward that the intersection of all compact convex sets containing the compact set is the convex hull of the compact set. But to prove that, it seems that one would have to also disprove that a convex hull that isn't compact can't exist. namely, that if it were the case, then that convex set could ultimately be shown to be closed and bounded, which would confirm that the intersection is in fact the convex hull. Any ideas on how to prove that such a set would be closed and bounded?