TITLE: For every rational number, does there exist a sequence of irrationals which converges to it?
QUESTION [14 upvotes]: I can think of of examples where a sequence of irrationals converges to $0$. But if we pick any rational will there always exist a sequence of irrationals which converges to it?
I cannot find a straight answer to this question.

REPLY [2 votes]: Yes. Consider
$\frac{p}{q} - \frac{\sqrt{2}}{n}$