Characterization of Paracompactness in T3 Space/Lemma 1

Theorem
Let $T = \struct{X, \tau}$ be a $T_3$ space.

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

Let:
 * $\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$

where $V^-$ denotes the closure of $V$ in $T$.

Then:
 * $\VV$ is an open cover of $T$

Proof
Let $x \in S$.

By definition of open cover:
 * $\exists U \in \UU : x \in U$

From Characterization of T3 Space:
 * $\exists V \in \tau : x \in V : V^- \subseteq U$

Hence:
 * $V \in \VV$

Since $x$ was arbitrary, $\VV$ is an open cover by definition.