The exercise is from Stein-Shakarchi's Real Analysis (Chapter 1, ex. 24).
Does there exist an enumeration $\{r_{n}\}_{n=1}^\infty$ of the rationals 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. [Hint: Find an enumeration where the only rationals outside of a fixed bounded interval take the form $r_n$, with $n=m^2$ for some integer $m$.]
While, I understand that we probably need some enumeration of the rationals such that the only rationals outside a fixed bounded interval are of the form $r_{m^{2}}$ for some $m$, I'm having trouble seeing how to get such an enumeration.
As always, help is very appreciated :)