Suppose that $f_n : [a,b] \to \mathbb{R}$ is a sequence of differentiable functions that is pointwise bounded. Assume in addition that $|f_n'(x)| \leq 1$ for all $n \geq 1$ and all $x \in [a,b]$. Prove that some subsequence of $(f_n)$ is uniformly convergent.

I'm not really sure where to start


Hint: By mean Value theorem We have $|\frac{f_n(x)-f_n(y)}{x-y}|=|f_n'(c_n)|<1$,try to show by the given data $f_n$s are uniformly bounded equicontinous, Now apply Arzela-Ascolli Theorem.

  • $\begingroup$ What about the condition that $(f_n)$ is uniformly bounded? $\endgroup$ – icaruss Feb 8 '13 at 12:38
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    $\begingroup$ @carl It also follows from the MVT that $|f_n(x)| \le |f_n'(c)||x| + |f_n(0)|$ which combines with the two conditions you are given to yield uniform boundedness. $\endgroup$ – user38355 Feb 8 '13 at 15:14

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