Does every line segment in $\mathbb{R}^n$ contains a point having only rational coordinates? Let $S$ be the subset of $\mathbb{R}^n$ consisting of all points which have only rational coordinates. I know that $S$ is a dense subset of $\mathbb{R}^n$. Is it true that every line segment in $\mathbb{R}^n$ intersects S?

No, of course not. Consider for example the line $y=\pi$ in $\mathbb{R}^2$.
