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Let $f_n$ be a sequence of differentiable functions on $[0,1]$ such that $|f_n'(x)| \le M$ (the absolute value of the derivative of $f_n$ at $x$) for all $n$ and for all $x$ in $[0,1]$. Show that $f_n$ has a uniformly convergent subsequence.

Partial solution: The $f_n$s are equicontinuous (by the mean value theorem). How to prove that the $f_n$s are pointwise bounded, so that we can use Arzela-Ascoli?

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    As the problem stands, you cannot; it may be false. For example, let $f_n(x)=n$. Is there anything else you know about $\{f_n\}$?2012-02-07

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