In my book's exercises section I am asked to prove that every bounded, open set in $\mathbb{R}$ is the union of disjoin open intervals.
Looking around the internet I have found many strategies that involve assigning each interval to a rational number. This has the effect of not only showing that the claim holds, but also showing that the intervals being union-ed are countable. As far as I have seen, they mostly proceed like this:
Figure that any member of a nonempty open subset $A$ of $\mathbb{R}$ is part of some open interval within $A$. By properties of intervals in $\mathbb{R}$, that interval contains a rational. With this reasoning, then, we can get an open interval corresponding to any point in $A$. Choose only the disjoint intervals. Since $\mathbb{Q}$ is countable, the number of intervals is also countable. The union of these disjoint intervals will be $A$.
Is there another way to do this that doesn't involve an extraneous set like $\mathbb{Q}$ and doesn't end up proving a stronger claim? More specifically, is there a way to solve it that depends on properties of bounded open sets (and possibly balls)?
If not, how would you deduce from the desired outcome that you would need to involve the subset $\mathbb{Q}$ ? It seems to come out of nowhere that you need to use its density properties in $\mathbb{R}$ to complete the proof.
Thank you!