Understanding the Proof of the Arzela-Ascoli Theorem from Carothers

Because if you have a uniformly convergent subsequence and the space is closed your subsequence will converge in the space, then you have a that for any sequence you can extract a convergent subsequence in the space, which is one of the characterizations of a compact space.

If the space is equicontinuous any sequence inside it is equicontinuous.

Same as above. Or: $$\{f_n\} \subseteq \{f\in\mathcal{F}\}$$

so

$$\sup_{f_n}\||f_n||_{\infty} \leq \sup_{f\in\mathcal{F}}||f||_{\infty}<\infty $$

For each $x$ he has some subsequence $f_{n_i}(x_i)$ that converges pointwise, taking the intersection of all the $n_i$ and renaming as $n$ he gets a sequence that converges for every $x_i$