Intersection of intervals in $\mathbb{R}$

Question

I have been searching for some question like that but hadn't found.

Let $I$ and $J$ be intervals in $\mathbb{R}$ such that $(I\cap \mathbb{Q})\cap(J \cap \mathbb{Q})$ = $\varnothing$

Prove: $I \cap J$ has at most one element.

I tried to assume negatively that there are two elements in $I \cap J$ - hoping I'll get they both equal finally. But I didn't succeed to find a way showing this.

Answer 1

Hint: if two elements are in an interval $I$, any element between them is also in $I$. Also, note that $\mathbb{Q}$ is dense in $\mathbb{R}$.