For a measure space prove if $f$ is measurable and bounded then there exists simple functions that converge to it?

Question

Let $(X,\Sigma)$ be a measure space and $f:X\rightarrow \mathbb{R}$ be measurable and bounded. Then there exists a sequence of simple measurable functions $f_n$ such that $f_n\rightarrow f$ uniformly, ie: $$\forall\epsilon>0 \mbox{ } \exists N \mbox{ } s.t. \forall x \in X \mbox{ } \forall n\geq N, |f_n(x)-f(x)|<\epsilon $$