This may be a non-standard question, and possibly irrelevant to the participants. Nevertheless, after reading the first chapter of Terence Tao's book on measure theory, I ended up thinking about what would be the limiting set of a set $\frac{\mathbb{N}}{N}\cap [1, 2]$, (where N is a positive integer) and then let $N \to \infty$.
i.e. $\lim_{N\to \infty} \frac{\mathbb{N}}{N}\cap [1, 2]$ where $\frac{\mathbb{N}}{N} = \{\frac{n}{N}: n\in\mathbb{N}\}$. At first superficial glance it seems like the limiting set is $\mathbb{Q}\cap[1,2]$, but maybe I'm missing some subtle details.
Edited - added image: This question was motivated by Tao's definition of a length of a Jordan measurable interval:
