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!