I'm working on the same problem as in this post. I understand the solutions provided in the answers. The question basically asks us to find an enumeration of the rationals $\{r_n\}_{n≥1}$ such that the complement of $\bigcup_{n=1}^{\infty}{\left(r_{n}-\frac{1}{n},r_{n}+\frac{1}{n}\right)}$ in $\mathbb{R}$ is non-empty.
I found the solutions rather complicated. What's wrong with my approach? For all $n \in \mathbb{N}$, let $r_n = 2^{n^2}$. This way, I know $[-\infty,1)$ is in the complement of $\bigcup_{n=1}^{\infty}{\left(r_{n}-\frac{1}{n},r_{n}+\frac{1}{n}\right)}$. Doesn't that complete the problem? What did I miss?