Proving convergence of Newton's method Consider the recursive sequence
$$ z_{n+1} = {1 \over 2}\left ( z_n + {1 \over z_n} \right )$$
where we start at some point $z_0 = x_0 + i y_0 \in \mathbb C$. This is Newton's formula to find the roots of $p(x) = x^2 -1$. By taking the limit on both sides we see that if the sequence converges then the limit must be either $1$ or $-1$. But:

How do you prove this sequence converges? Of course this is the very
  first thing we must show, even before calculating the limit!

Normally I would show it's a Cauchy sequence but I don't see how to do that here. 

Assume $z_1 > 0$, first you prove that $z_n \geq 1, \forall n \geq 2$, and this is easily done by apply AM-GM inequality to the right side of the equation. Next, $0\leq z_{n+1} - 1 = \dfrac{(z_n-1)^2}{2z_n}=(z_n-1)\dfrac{z_n-1}{2z_n}< \dfrac{z_n-1}{2}<...< \dfrac{z_1-1}{2^n}$. Thus by squeeze theorem, $z_n \to 1$.
