Prove the following limit using an $\epsilon - \delta$ proof for complex numbers

Question

Prove $\lim \limits_{z \rightarrow i} (z^2) = -1$ using an $\epsilon - \delta$ proof. I am unsure of how to even begin.

Answer 1

Try and solve $$ |z^2-(-1)|<\varepsilon $$ taking into account that $z^2+1=(z-i)(z+i)$ and that, provided that $|z|<1$, $|z+i|\le|z|+|i|<2$.

It's essentially the same as proving that $\lim_{x\to2}x^2=4$ in the real numbers.