If $\{f_{n}\}_{n\geq 1}$ is a sequence of continuous functions on $\mathbb R$, with $|f_{n}(x)|\leq 1$ for all $x\in \mathbb{R}$, and all $n\geq 1$. Does there exist a subsequence which converges uniformly (or pointwise) to some continuous function $f$?
As I know the Arzelà –Ascoli theorem works for closed intervals $[a,b]$, I don't know if there is something in case of $\mathbb{R}$?
EDIT: If this assumption help we can consider it: the sequence $\{f'_{n}\}$ is uniformly bounded on $\mathbb R$.