Let $f(x)$ be $x^2-x-2$. I want to find the root using FPI in an interval where it will converge. I have chosen $g(x)=x^2-2$ and so $g'(x)=2x$. The convergence condition, $|g'(x)|<1$ is obviously satisfied in $-0.5<x<0.5$.
Problem: I have failed to find a consistent interval outside of this convergence interval up to $\pm 1$ for which the iteration consistently diverges.
Question: Does this convergence condition only guarantee convergence (in the bounds) but not divergence (outside the bounds)?