Union of arbitrary neighborhoods of rational numbers

Let $Q$ be the set of rational numbers, which we enumerate to write as $\{ r_n \}_{n=1}^\infty$.

For each $r_n$, let us allocate some $\epsilon_n >0$ and consider the interval $[r_n -\epsilon_n, r_n]$.

Then, my question is

Is it always true that $\cup_n [r_n -\epsilon_n, r_n] = \mathbb{R}$? Or is it possible to find $\epsilon_n$'s such that $\cup_n [r_n -\epsilon_n, r_n] \subsetneq \mathbb{R}$?

This seems like a tricky question for me. Could anyone please help?