# Uniformly convergent subsequence of a uniformly bounded family of functions

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?

Let $$x. Then, there is a $$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.