Currently I am struggling to understand a solution to the following exercise:
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 $\lim_{n \to \infty} \ z_n = 1$.
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We notice that the only possible values for a limit would be $\pm 1$. We further observe that since $z_0$ is in the right half plane all $z_n$ are in there too. Next we observe the sequence $w_n$ defined by
$$ w_n := \frac{z_n-1}{z_n+1}$$
It holds that $w_{n+1} = w_n^2$. And since $|w_n| < 1 $ we see that $w_n$ converges to $0$. From $|z_n+1| \ge 1$ we deduce that $z_n$ converges to $1$.
I do not understand the part
From $|z_n+1| \ge 1$ we deduce that $z_n$ converges to $1$.
Could you explain that to me ?