Let $a>1$. We are given the following iterations: $$y_{k+1} = \frac{1}{2} (y_k + z_k^{-1})$$ $$z_{k+1} = \frac{1}{2} (z_k + y_k^{-1})$$ with $z_0 = 1$ and $y_0 = a$.
I need to show that $\{ y_k \}$ converges to $\sqrt{a}$ and $\{ z_k \}$ converges to $\frac{1}{\sqrt{a}}$ and then find the rate of convergence of $\{ y_k \}$.
By working with both formulas I got to $\frac{y_{k+1}}{z_{k+1}} = \frac{y_{k}}{z_{k}}$, so if I denote that $t_k = \frac{y_{k}}{z_{k}}$ I will have $t_{k+1} = t_k$, meaning the iterations $t_{k+1} = g(t_k)$ where $g(t) = t$.
But then $g'(t) = 1$, and I can't use theorems about convergence.
Help would be appreciated.
Well done to find $y_k/z_k$ is constant.
By induction, $$\frac{y_k}{z_k}=\frac{y_0}{z_0}=a$$ so replace $z_k$ by $y_k/a$ in the first equation, and you have a single, simpler iteration.