1
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What about the condition that $(f_n)$ is uniformly bounded? $\endgroup$ – icaruss Feb 8 '13 at 12:38
  • 1
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.