How to write a proof for this?

Question

Prove or disprove that if $A\subset [0, 1]$ and $m(A) > 0$, then there are $x$ and $y$ in $A$ such that:

$$|x− y| $$

Is an irrational number.

I know this is true as if there were only rationals then measure would be zero but how do I prove it rigorously ? How do I write it down ?

Answer 1

Hint: Fix $x \in A$. If $|x-y|$ is rational for every $y \in A$ what can we say about the cardinality of $A$?