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I am trying to solve the following problem:

Suppose $\{f_n\}$ is a sequence of functions that are continuous and differentiable on $[a,b]$. Suppose that the sequence $\{f_n\}$ and $\{f_n'\}$ are uniformly bounded on $[a,b]$. Prove that $\{f_n\}$ has a uniformly convergent subsequence.

I tried to apply Arzela-Ascoli theorem because $\{f_n\}$ is continuous on a compact set. Moreover, ${f_n}$ is pointwise bounded, but I got stuck proving $\{f_n\}$ is equicontinuous on $K$. I tried to use the Fundamental Theorem of Calculus, but I do not how to prove $f_n'$ is Riemann integrable. Any thoughts?

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Let $x<y$. Then, there is a $c\in (0,1)$ such that $|f_n(x)-f_n(y)|\le |f_n'(c)(x-y)|$ and this is less than $M|x-y|$ where $M=\sup \{f_n'(x):n\in \mathbb N;\ x\in [0,1]\}$ so $\{f_n\}$ is equicontinuous and the result follows by Arzela-Ascoli.

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