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?