a.e. convergence of uniformly bounded continuous functions to a bounded measurable function

Question

I would like to show that for a given function $f\in B[a,b]$ there exists a sequence $\{f_n\}_{n\in\mathbb{N}}\subset C_c[a,b]$ such that $$ f_n\to f \ a.e.\ \text{as }n\to\infty\quad \& \quad\sup_n||f_n||_\infty<\infty, $$ where $B[a,b]$ is the space of real-valued bounded Borel measurable functions on the interval $[a,b]$ and $C_c[a,b]$ is the space of real-valued compactly supported continuous functions on $[a,b]$.