Could someone please check my work

Prove that the intersection of any collection of compact sets is compact. Let { An : n E N} be an arbitrary collection of compact sets, so that An is the arbitrary n = N intersection of any collection of compact sets. By Theorem 3.5.5 (Heine Borel), since every An in the collection is compact, then every An is bounded and closed. By Corollary 3.4.11, the intersection of any collection of these An closed sets is closed, so n An is closed. n = N Since every subset of a bounded set is bounded, then since An is bounded and n An C An n = N then An is bounded. n = N Since An is closed and bounded, then it is compact. n = N Therefore, the intersection of any collection of compact sets An is compact. n = N