A Random Line Intersecting the Corner of a Square on an Infinite Grid of Squares

Question

I thought that I could consider the grid as the analytic plane and the corners as $x,y \in \mathbb{Z}$ and the line as $y = mx + n$ for any arbitrary pair of $m$ and $n$. Then if $m$ is rational, the line would intersect infinitely many corners, otherwise it would at most intersect once. I have two questions. First, is my reasoning correct? Second, is a pure geometrical solution possible without using the properties of numbers and considering corners as points on a coordinate system?