Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 1

Theorem
Let $T = \struct{X, \tau}$ be a topological Space.

Let $\VV = \ds \bigcup_{n = 0}^\infty \VV_n$ be an cover of $X$, where each $\VV_n$ is a set of subsets of $X$ for each $n \in \N$.

For each $n \in \N$, let:
 * $W_n = \ds \bigcup \VV_n$

Let:
 * $\WW = \set{W_n : n \in \N}$

Then:
 * $\WW$ is a cover of $X$

Proof
Let $x \in X$.

By definition of cover of set:
 * $\exists V \in \VV : x \in V$

By definition of union:
 * $\exists n \in \N : V \in \VV_n$

From Set is Subset of Union:
 * $V \subseteq W_n$

By definition of subset:
 * $x \in W_n$

It follows by definition, $\WW$ is a cover of $X$.