When applying the Arzela Ascoli theorem, I am little bit confused as to what needs to be done to show that a set of functions in $C_0([a,b])$ is closed.
Sifting through MSE I found a claim that :
"a set of functions is closed if it contains the pointwise limit and converges uniformly to it"
Is this statement true? What is the motivation behind it and can someone provide a reference to or a proof of the above?
A subset $C$ of a normed space $X$ is closed iff convergent sequences of elements of $C$ have their limits in $C$. In other words, convergent sequences in $C$ cannot escape $C$ (hence, the name closed.)
In the case of $C[a,b]$, a sequence $\{f_n \}\subset C[a,b]$ converges to $f\in C[a,b]$ iff $$ \lim_{n\rightarrow\infty} \|f-f_n\|=\lim_{n\rightarrow\infty}\sup_{x\in[a,b]}|f(x)-f_n(x)| = 0. $$ This is equivalent to uniform convergence of $\{ f_n\}$ to $f$.
If you have an equicontinuous set $\mathcal{E}$ of functions in $C[a,b]$, then pointwise convergence of a sequence $\{ f_n \}\subseteq\mathcal{E}$ implies uniform convergence.