For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$. How to prove that for every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n \rightarrow a$.

By Riemann's series theorem it follows, that for every $x \in \mathbb{R}$ there is an arrangement $\sigma:\mathbb{N} \to \mathbb{N}$ such that $\sum_{n=1}^\infty \frac{(-1)^{\sigma(n+1)}}{\sigma(n)} = x$. Note that $b_n := \sum_{k=1}^n \frac{(-1)^{\sigma(n+1)}}{\sigma(n)}$ is a rational sequence.
