Characterization of Paracompactness in T3 Space/Statement 2 implies Statement 3

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

If every open cover of $T$ has a locally finite refinement then:
 * every open cover of $T$ has a closed locally finite refinement

Proof
Let every open cover of $T$ have a locally finite refinement.

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 $T$.

Lemma 1
By assumption:
 * there exists a locally finite refinement $\AA$ of $\VV$.

Let:
 * $\BB = \set{A^- : A \in \AA}$

From Closures of Elements of Locally Finite Set is Locally Finite:
 * $\BB$ is locally finite

Lemma 3
Since $\UU$ was an arbitrary open cover of $T$ it follows that:
 * every open cover of $T$ has a closed locally finite refinement.