Let $p$ be a complex number. Let $ z_0 = p $ and, for $ n \geq 1 $, define $z_{n+1} = \frac{1}{2} ( z_n - \frac{1}{z_n}) $ if $z_n \neq 0 $. Prove the following:
i) If $ \{ z_n \} $ converges to a limit $a$, then $a^2 + 1 = 0 $
ii) If $ p $ is real, then $ \{ z_n \} $, if defined, does not converge
iii) If $ p = iq $, where $ q \in \mathbb{R} \backslash \{0\} $, then $ \{ z_n \} $ converges.
I have been able to do the first two parts of this (the second is because the sequence would be real, but would have to have a complex limit). I am stuck on the third part, though. Any help would be greatly appreciated.
Thanks