Sigma-Locally Finite Cover has Locally Finite Refinement

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

Let $\SS = \ds \bigcup_{n = 0}^\infty \SS_n$ be a $\sigma$-locally finite cover of $X$, where each $\SS_n$ is locally finite for all $n \in \N$.

Then:
 * there exists a locally finite refinement $\AA$ of $\SS$

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

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

Lemma 1
For each $n \in \N$, let:
 * $C_n = B_n \setminus \ds \bigcup_{i < n} B_i$

Let:
 * $\CC = \set{C_n : n \in \N}$

Lemma 2
From Sigma-Locally Finite Cover and Countable Locally Finite Cover have Common Locally Finite Refinement:
 * there exists a common locally finite refinement $\AA$ of both $\CC$ and $\SS$