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 $\SS$