In real analysis there was an easy property that converted limits to infinity in limits at zero. More precisely, $1/z_n$ converges to 0 if and only if $z_n$ diverges (this is converges to infinity).
I want to apply this property to complex numbers as follows:
Let $z_n$ a complex sequence. $1/z_n$ converges to 0 if and only if $z_n$ diverges
I've written $1/z_n = x_n/(x_n^2+y_n^2)-i(y_n/(x_n^2+y_n^2))$ where $x_n = Re(z_n)$ and $y_n = Im(z_n)$ but I can't see either of the implications.