ILL 1990 Problem 68

In coordinate plane, a variable point $M$ , starting from the origin $O(0, 0)$ , moves on the line $l$ with slope $k$ , where $k$ is an irrational number.(i) Prove that point $O(0, 0)$ is the only rational point (namely, the coordinates of which are both rationals) on the line $l.$ (ii) Prove that for any number $\varepsilon > 0$ , there exist integers $m, n$ such that the distance between $l$ and the point $(m, n)$ is less than $\varepsilon.$