every complex number is within $\tfrac{\sqrt 2}{2}$ units of a Gaussian integer

Question

From Wikipedia it says that "It is easy to see graphically that every complex number is within $\frac{\sqrt 2}{2}$ units of a Gaussian integer." How do I go about seeing this?

Answer 1

For every real number $t$ there is an integer $n$ such that $|t-n|\le \frac12$: just take $n$ to be the integer nearest $t$.

Apply this to $z=x+yi$ and get $j=m+ni$ with $|x-m|\le \frac12$ and $|y-n|\le \frac12$.

This implies that $|z-j|^2 = |x-m|^2 + |y-n|^2 \le \frac12$.