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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 $f$.2012-06-28

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