For every rational number, does there exist a sequence of irrationals which converges to it?
Solution 1:
Assume your number is $\frac{p}{q}$. Then the sequence $$a_n=\frac{\pi}{n}+\frac{p}{q}$$ converges to the given number and is irrational (any irrational number in the place of $\pi$ would do).
Solution 2:
Yes, take a sequence consisting of your sequence of irrationals converging to $0$ plus your desired rational limit.
Solution 3:
Yes: If $r\in\Bbb Q$, then $\forall n\in\Bbb N$: ${rn\over n+\sqrt2}\in{\Bbb Q}^c$ and $$\lim_{n\to\infty}{rn\over n+\sqrt2}=r.$$