[Math] Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Question

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then
let $V$ be an open cover of $A \cup B$. Then let $A$ be compact, then $ V$ has a finite subcover of $A$ . But suppose $B$ is not compact. Then $V$ does not contain a finite subcover of $B$. Thus their union will not be have a finite subcover of $ V$ that converse $A \cup B$.

Can someone please help me? I already proved if $A, B$ are compact , then their union is compact. So I was trying to use a similar argument.
Any feedback/help would really help. Thanks

Answer 1

There is a problem with your contrapositive.

Hint: show that a closed part of a compact set is compact.