Fully Normal Space is Paracompact

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

Then $T$ is paracompact.

Proof
From the definition, $T$ is fully normal :
 * $T$ is fully $T_4$
 * $T$ is a $T_1$ (Fréchet) space.

Then $T$ is fully $T_4$ every open cover of $S$ has a star refinement.

Let $\UU$ be an open cover for $T$.

Then from the definition, there exists a cover $\VV$ for $T$ such that:
 * $\ds \forall x \in S: \exists U \in \UU: \paren {\bigcup \set {V \in \VV: x \in V} } \subseteq U$

Recall from the definition of paracompact:
 * $T$ is paracompact every open cover of $S$ has an open refinement which is locally finite.