Intersection of Compact sets is compact

Question

If $X$ is Hausdorff, and $\{C_\alpha\}_{\alpha\in A} $is a collection of sets that are compact in $X$, then $\bigcap_{\alpha \in A}C_\alpha$ is compact in X. I know the proof to the statement should be easy, but I am stuck at how I could use the condition that $X$ is Hausdorff. Can anyone give me some hints?

Answer 1

Hint: closed subsets of a compact (sub)space are compact