Take a sheet of grid paper and draw a straight line in any direction from the origin. What is the closest non-zero grid point $\boldsymbol{p}\in\mathbb{Z}^2$ within a distance $\epsilon>0$ of the line?

I have been unable to construct an example where the distance is large for any $\epsilon$. I would like to prove the following:

$\forall \epsilon>0 ~~ 
\exists R>0 ~~
\forall \boldsymbol{v} \in \mathbb{R}^2 ~~ 
\exists\boldsymbol{q}\in\mathbb{Z}^2, \|\boldsymbol{q}\| < R$ such that:

$\frac{|p_x v_y - p_y v_x|}{\|\boldsymbol{v}\|}<\epsilon$

where the LHS is the point-line distance.