Given: $g(x)=\frac{1}{2}(x+\frac{a}{x})$ for $a\in \mathbb R_{>0}$
Question: For which starting values $x_0>0$ does the iteration $x_{k+1}=g(x_k)$ converges?
My thoughts: Should I find an interval $I\subset \mathbb R$ such that $g$ has lipschitz constant smaller than $1$ on $I$?
Are there other ways to solve this?