[Math] bounded measurable function is the uniform limit of a sequence of simple functions

Question

Let $ f: \mathbb R \to \mathbb R $ a non-negative bounded measurable function. Prove that there exists a sequence of simple non-negative functions $ (f_n)_{n \in \mathbb N} $ such that $ f_n \to f$ uniform.

Searching on Wikipedia I found the following http://en.wikipedia.org/wiki/Simple_function

but I can't understand why the converge is uniform.

Any help?

Answer 1

Hint: if $A \le f \le B$, split up $[A,B]$ into $n$ equal intervals and choose a value for $f_n$ in each one.