GRE subject exam 0568 Q.20

Question

This is the question with the answer:

I do not understand why always there exist a sequence of rationals and a sequence of irrationals that tends to $y$ as $n$ tends to $\infty$, on what mathematical fact the author depends in using this idea, could anyone explain this for me?

Answer 1

This is true because the set of rational numbers is a dense subset of $\mathbb{R}$. Similarly for the set of irrational numbers.