TITLE: Global convergence for Newton's method in one dimension
QUESTION [6 upvotes]: I'm looking for a Theorem that I can cite which proves that Newton's method for finding a zero of a function converges globally and quadratically if the function $f : [a, b] \rightarrow \mathbb{R}$ is increasing and convex and has a zero $r \in [a,b]$ with $f(r) = 0$ and $f'(r) \neq 0$, with starting point $x_0 \geq r$. I couldn't find a formal Theorem in any book I have and also not online. 
It probably isn't too hard to prove, but due to space restrictions I would like to cite an existing result and I would like to avoid writing "it is a well-known result..." or "it can be easily shown that...".

REPLY [2 votes]: The global theorem your are asking for, i.e., monotone convergence for increasing convex functions with a zero, generalizes to the $n$-dimensional case and can be found as Theorem 13.3.7., p. 453, in J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, 1970. By the local results, the iterates will eventually converge quadratically.