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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?

samantha
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    Draw the lattice $\mathbb{Z}[i]$ in the complex plane. Then it is obvious. What is the diagonal length of the unit square ? – Dietrich Burde Feb 19 '16 at 11:21

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

lhf
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