Compact Hausdorff Space is T4

Theorem
Let $T = \struct {S, \tau}$ be a compact Hausdorff space.

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 Subspace of 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 $S$ are separated by neighborhoods.

Hence the result.