Let us consider $C(\mathbb{R})$, the space of continuous functions on the reals.
Q. Does there exist a sequence $\{f_n\}$ in $C(\mathbb{R})$ such that for every $f\in C(\mathbb{R})$ one may find a subsequence $\{f_{n_k}\}$ of $\{f_n\}$ pointwise converging to $f$, i.e., $f(x)=\lim f_{n_k}(x)$ for all $x$?