In Ridgway Scott's Numerical Analysis, the following two problems appear:
In the first exercise, there was a clear mention of $x_0$ being sufficiently close to $\alpha$. So I knew that I should be proving local convergence, not global. However for the second exercise, there is no such mention, so I assumed I should be showing global convergence. The author includes a solution to the exercise at the end of the section, and in the solution uses the following argument:
My question is, does this not assume that $x_0$ is starting sufficiently close to $\alpha$? If I'm not mistaken, this solution uses that $|g'(\alpha)|<1$ to find a Lipschitz constant $\lambda<1$ and obtain (2.67). But the function $g$ is not Lipschitz-continuous over the entire domain, only on a small interval around $\alpha$, right?
If this is the case, then this solution only proves local convergence? Does the convergence hold globally at all, and if so how is it proven?