Compact Hausdorff spaces are normal

Statement

Let $X$ be a compact space that is Hausdorff. Then $X$ is normal.

Remarks

Proof sketch: closed subsets of $X$ are compact, by closed subsets of a compact set are compact. In a Hausdorff space, a point and a disjoint compact set have disjoint open neighbourhoods. A second use of compactness separates two disjoint compact sets. With Urysohn's lemma, this shows that compact Hausdorff spaces have many continuous real functions.

Explanations

L1 human

In a compact Hausdorff space, you first separate points by open sets, and then compactness lets you combine finitely many of these choices. The result separates any two disjoint closed sets.