Compact Space is Paracompact

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

Then $T$ is paracompact.

Proof
From the definition, $T$ is compact every open cover of $S$ has a finite subcover.

From Subcover is Refinement of Cover, it follows that every open cover of $S$ has an open refinement which is locally finite.

This is precisely the definition of paracompact.