Prove fixed point iteration of $g(x) = \frac{1}{2}x + \frac{A}{2x}$ converges to $\sqrt{A}$ If we define, for any positive number $A$:
\begin{align*}
  g(x) &= \frac{1}{2}x + \frac{A}{2x} \\
  g'(x) &= \frac{1}{2} - \frac{A}{2x^2} \\
\end{align*}
And we define the normal fixed point iteration sequence $\{ p_n \}_{n=0}^\infty$ based on $p_n = g(p_{n-1}), n \ge 1$
How can we prove that this will converge to $\sqrt{A}$ when $p_0 > 0$? What happens if $p_0 < 0$?
The normal textbook proof of fixed point convergence depends on $g'(x) \le k < 1$ which is clearly not the case here.

I am not sure that this could be an answer.
$$g(x) = \frac{1}{2}x + \frac{A}{2x}=x -\frac{x^2-A}{2x}$$
$$x_{n+1} = x_n -\frac{x_n^2-A}{2x_n}$$ is the Newton formula for solving $x^2-A=0$
