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

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

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

For each $n \in \N$, let:
 * $B_n = \ds \bigcup \SS_n$

Let:
 * $\BB = \set{B_n : n \in \N}$

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

Proof
Let $x \in X$.

By definition of cover of set:
 * $\exists S \in \SS : x \in S$

By definition of union:
 * $\exists n \in \N : S \in \SS_n$

From Set is Subset of Union:
 * $S \subseteq B_n$

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

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