Use of Arzela-Ascoli theorem to ensure that there is a subsequence of a sequence of functions that converges to an infinitely differentiable function.

Question

I'm trying understand how to prove this result. I read the answer given on the post, but there are three points of the answer that I didn't understand:

1. Why $f'_{n_{k_{i}}} \rightarrow f'?$ I know that $f'_{n_{k_{i}}} \rightarrow g$ for some function $g$ by Arzela-Ascoli's theorem, but how exactly can I ensure that $g = f'?$

2. How ensure that exists $f'?$ Because the answer only ensure that $f$ is continuous and I don't think that Mean Value Theorem can be ensure that $f \in \mathcal{C}^1$.

   3. Why is necessary to use the Cantor's diagonal argument? As the author of the answer said, is the process of choosing subsequences no longer enough to ensure the function's differentiability?

Thanks in advance!

Answer 1

Suppose $f_n \to f$ uniformly and $f_n' \to g$ uniformly. Then $f_n(x)-f_n(x_0)=\int_{x_0}^{x} f_n'(t) \, dt$. Take limits to get $f(x)-f(x_0)=\int_{x_0}^{x} g(t)\, dt$. This implies that $f$ is differentiable and $f'=g$. Repeated use of this theorem answers all your questions.