Oftentimes in density arguments we let $\{x_n\}$ be a dense sequence and this is sufficient to imply the desired result.

From a research question I am working on I have simplified the example/counterexample to the following problem, which I believe is perhaps a nice exercise in choice (and yet, I cannot make a good one).

Precisely, can one choose a "good" ordering of a dense set $\{x_n\} \subset (-1,1)$ and a "good" sequence $r_n>0$ such that

$\sum_n r_n <\infty$

and

$|\{x \in (-1,1):x\in B(x_n,r_n) \text{ for infinitely many } n\}|>0$

where I have used $|\cdot|$ to denote the Lebesgue measure on $\mathbb{R}$.