As an example of Arzela-Ascoli theorem in intro. to topology course, the following case was exhibited:
Let $A\subset C[a,b]$ be a set of differentiable functions such that: $\forall f\in A,\ \vert f \vert \leq M_1,\ \vert f' \vert \leq M_2$
thus A is bounded and equicontinuous, thus every sequence $\{f_n\}$ has a converging subsequence.
My question is: The limit function isn't necessarily in $A$, right?
A isn't necessarily a closed set, and the more detaild chain of consequences is: bounded & equicontinuous $\rightarrow$ absolutely bounded $\rightarrow\ \bar A$ is closed and absolutely bounded $\rightarrow\ \bar A$ is compact $\rightarrow$ every sequence $\{f_n\}\subset \bar A$ has a converging sub-sequence.
So a sequence in $A$ might converge to $C[a,b]\supset \bar A \ni f \notin A$, am I right?
As pointed out in the comments by zhw and Fimpellizieri:
We only know that $A$ is precompact for the metric induced by the uniform norm, but it may not be closed. For example, if $A=\left\{ x\mapsto 1/n,n\in\mathbb N, n\geqslant 1\right\}$, the limit of $\left(1/n\right)$ does not belong to $A$.