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This question already has an answer here:

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?

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marked as duplicate by José Carlos Santos real-analysis May 9 at 16:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Hint:

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

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