# bounded measurable function is the uniform limit of a sequence of simple functions

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?

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The key point behind the uniform part of this statement lies in the boundedness of $f$. – T. Eskin Jun 28 '12 at 18:53

## 1 Answer

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.

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