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

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 $$n_1. Then $$p_{n_k}\to f$$ pointwise on $$\mathbb R$$. (Moreover, this convergence is locally uniform.)