Show that for any two real numbers $x, y \in [0,1]$, there exist $a,b \in \mathbb{N}^0$, $a,b \leq 18$ not both equal to $0$ such that $|ax-by|<1/9.$

Question

This is a problem coming from a math gazette. I have tried to approach this problem by dividing the interval $[0,1]$ into intervals $\left[0, \frac{1}{9}\right], \left[\frac{1}{9}, \frac{2}{9}\right], \ldots , \left[\frac{8}{9}, 1\right]$ and seeing where $x$ and $y$ could be and choosing $a$ and $b$ accordingly, but I didn't end up with anything close to a solution. Any help will be appreciated.