Let $A=\mathbb{Z}[\sqrt{-2}]$ and let $a,b \in A,$ $b \neq0$. Set $\gamma := \dfrac{a}{b}$. I am struggling to show that there exists $c \in A$ such that $|c-\gamma|\leqslant \dfrac{{\sqrt{3}}}{2}$.
I would like to use this fact to then prove that $A$ is a euclidean domain.
Any help would be appreciated, thank you.
Every complex number $\gamma$ is inside some rectangle with length $1$, height $\sqrt2$, and corners in ${\bf Z}(\sqrt{-2})$. So $\gamma$ is within $\sqrt3/2$ of one of those corners.