I am trying to show that if $A$ and $B$ are compact in a Hausdorff space, then their intersection is compact.
The way I did it is as follows. $A$ and $B$ are compact in a Hausdorff space, so they are closed. As a consequence, their intersection (denote it by $C$) is closed. But $C$ is closed in $A$, which is compact, so $C$ is compact.
I am pretty sure this is correct, but so far compactness seems a bit tricky, so I want to ensure I get everything right.