In class we looked at the following exercise:
Let $a \ne 0$ be a complex number and let the sequence $(z_n)_n$ be recursively defined as
$$z_{n+1} = \frac{1}{2} \left( z_n+\frac{a}{z_n} \right)$$
for $n \ge 0$. For which $z_0 \in \mathbb{C}$ is the above sequence well defined and if it is, what is it's limit?
Someone said that it suffices to reduce the above exercise to the exercise below:
Let $z_0 = x_0+iy_0 \ne 0$ be a complex number and let the sequence $(z_n)_n$ be recursively defined as
$$z_{n+1} = \frac{1}{2} \left( z_n+\frac{1}{z_n} \right)$$
for $n \ge 0$. Show that if
$x_{0} > 0$ then $\lim_{n \to \infty} \ z_n = 1$.
$x_{0} < 0$ then $\lim_{n \to \infty} \ z_n = -1$.
$x_{0} = 0$ and $y_{0} \ne 0$ then $(z_n)_n$ is not defined or divergent.
I do not understand how such a reduction should work. Could you please explain that to me?
Note: Here I posted major parts of my solution to the second exercise.