Fully Normal Space is Paracompact

Theorem
Let $T = \left({S, \tau}\right)$ 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 $\mathcal U$ be an open cover for $T$.

Then from the definition, there exists a be a cover $\mathcal V$ for $T$ such that:
 * $\displaystyle \forall x \in S: \exists U \in \mathcal U: \left({\bigcup \left\{{V \in \mathcal V: x \in V}\right\} }\right) \subseteq U$

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