How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x

Question

How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x<y$?

Sorry, but I don't even know how to start. Any ideas and impulses?

Answer 1

Hint:

Find a nonzero rational $r$ such that $x\sqrt 2<r<y\sqrt2$