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 $x$?



Some special sequence in $C(\mathbb{R})$
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.)


