Could someone please check my work

Prove that the intersection of any collection of compact sets is compact. ' .511 Suppose A1 and A2 are collections of compact subsets of R . e_ compacts sets is closed by the same corollary. '1 1 Since A1 is bounded and A1 0 A2 Q A1, then since every subset of a bounded set is bound ' By Theorem 14.5 (Heine Borel), since A1 and A2 are compact, then they are closed and bounded. .- .w.-..,.c., ....__v m...\" v\"..a_..___.._._~..__.. By Corollary 3.4.11, the intersection of the collection of closed sets, A1 and A2 is closed. Since A1 and A2 are arbitrary collections of compact sets, the intersection of any collection of then A1 (1 A2 is bounded. Since A1 and A2 are arbitrary collections of compact sets, then the intersection of any of compact sets is bounded.' Since the intersection of any collection of compact sets is closed and bounded, ,. : 14.5 (Heine Borel), the intersection of any collection of compact sets is compa Therefore, the intersection of any collection of compact sets is compact.\" J\" ,7