Compact Hausdorff Space is T4

Theorem
Let $T = \left({X, \vartheta}\right)$ be a Hausdorff space which is compact.

Then $T$ is a $T_4$ space.

Proof
We have that a Compact Subspace of Hausdorff Space is Closed.

We also have that a closed subset of a compact space is compact.

We also have that Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods.

$T$ is a $T_4$ space when any two disjoint closed subsets of $X$ are separated by neighborhoods.

Hence the result.