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 $x$?
Yes. Take the countable set $P$ of all polynomials with rational coefficients and enumerate it somehow so that $P=\{p_1,p_2,\dots\}$.
Given any $f\in C(\mathbb R)$, for each natural $k$ there exists some natural $n_k$ such that $\sup_{x\in[-k,k]}|p_{n_k}(x)-f(x)|<1/k$. Here without loss of generality we may assume that $n_1<n_2<\cdots$. Then $p_{n_k}\to f$ pointwise on $\mathbb R$. (Moreover, this convergence is locally uniform.)
