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I am trying to prove the following variation of the Arzel-Ascoli theorem:

Let $(f_n)n$ be a sequence of differentiable functions on $[a,b]$ whose derivatives are uniformly bounded and there is an $x_0 \in [a,b]$ such that $(f_n(x_0))_n$ is bounded in $R$. Prove that $f_n$ has a uniformly convergent subsequence. k I can prove that the sequence $f_n$ is equicontinuous but I am not sure how to use the fact that it is bounded at one point. It mean it is just a sequence of real numbers which is bounded thus we can find a converging subsequence $f_{n_k}(x_0)$ but how to continue for the uniform convergence part?

I would appreciate some help.

Thank you!

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  • $\begingroup$ The derivatives are uniformly bounded, and the sequence is bounded in one point - can you use it to show the whole family is uniformly bounded on the interval? $\endgroup$ – Marcin Łoś Feb 23 '14 at 18:47
  • $\begingroup$ I think I can show the fam. is pointwise bounded. I don't see how to prove it is unif. bounded. $\endgroup$ – Whats My Name Feb 23 '14 at 19:16
  • $\begingroup$ Hint: use mean value theorem and uniform bound on the derivatives. $\endgroup$ – Marcin Łoś Feb 23 '14 at 19:18
  • $\begingroup$ I can prove that using the MVT we have that the sequence is equicontinuous and point-wise bounded thus exactly the conditions we need to apply AA Theorem. $\endgroup$ – Whats My Name Feb 23 '14 at 19:37
  • $\begingroup$ Yes, exactly :) (I assume you mean "uniformly bounded") $\endgroup$ – Marcin Łoś Feb 23 '14 at 19:39

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