GRE subject exam 0568 Q.20

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?

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

• what about y is it rational or irrational or real? Jun 15 '17 at 11:27
• y is a real number. Jun 15 '17 at 11:43