The set $S$ of all rational numbers whose denominators are prime.
$S = \{(-1)^{n}+\frac{1}{n}\mid n\in\mathbb{N}\}$
$S = (0,1) $
I have answers for these from the back of a book but I'm not sure about the intuition behind finding the limit points.
Edit: I understand #3 now, all $x\in[0,1]$ are accumulation points since an open ball around any of those points contains infinitely many points of $S$.
Not sure on how to approach #1 though.