Problem Statement: Let $X$ be a $\sigma$-compact metric space, $\mathcal{F}\subset C(X;E)$ a family of continuous functions. If $\mathcal{F}$ is equicontinuous at each point of $X$ and $\mathcal{F}(x)$ is precompact in $E$ (Banach) for each $x\in X$, then any sequence of functions in $\mathcal{F}$ has a subsequence converging uniformly on compact sets (to a function $f\in C(X;E)$).
I'm really having trouble approaching this problem. I was told to use the Cantor-Tychonof diagonal argument, but I am not sure if I am applying it correctly.
By hypothesis: Since $X$ is $\sigma$-compact, $X=\bigcup _{i\geq 1} K_{i}$, with each $K_{i}\subset X$ compact, and $K_{i}\subset K_{i+1}$.
Since $\mathcal{F}$ is equicontinuous at each point of $X$, then taking $\epsilon >0$, we have $\forall x_{0}\in X$, $\exists \delta=\delta(\epsilon,x_{0})>0$ such that $$d(x,x_{0})<\delta\ \Rightarrow\ \lVert f(x)-f(x_{0})\rVert <\epsilon,\ \forall x\in X, f\in \mathcal{F}.$$ Since $\mathcal{F}(X)$ is bounded in $E$ for each $x\in X$, then $\exists M>0$ such that $$\lVert f(x)\rVert \leq M,\ \forall f\in\mathcal{F},\ x\in X.$$
So I want to consider some sequence $\left\{f_{n}\right\}_{n\geq 1}$ in $\mathcal{F}$ and a countable set $\left\{x_{1}, x_{2}, x_{3}, ...\right\}\subset X$, where each $x_{i}\in K_{i}$
Then we know $\left\{f_{n}(x_{1})\right\}_{n\geq 1}$ is equicontinuous and bounded. Then since $x_{1}\in K_{1}$, which is compact, the image $\left\{f_{n}(x_{1})\right\}_{n\geq 1}$ will be compact since $f_{n}$ is continuous. So there is some $N_{1}\subset \mathbb{N}$ so that $\left\{f_{n}(x_{1})\right\}_{n\in N_{1}}$ converges to some $f(x_{1})$? Continuing in this fashion, $\left\{f_{n}(x_{i})\right\}_{n\geq 1}$ will be compact since $f_{n}$ is continuous. So there is some $N_{i}\subset N_{i-1}$ so that $\left\{f_{n}(x_{i})\right\}_{n\in N_{i}}$ converges to some $f(x_{i})$.
Is this the right approach? Then once I define these limits $f(x_{i})$, I can construct the set $N_{0}$, where the $i$th element is the $i$th element of $N_{i}$? To conclude, do I just need to use the definition of uniform convergence on compact sets? I am not completely sure how to conclude here...
Edit: I found an error in my Problem statement, the assumption should be that $\mathcal{F}(x)$ is precompact in $E$, rather than bounded. I believe I can finish the proof now, using Arzela-Ascoli.