Global convergence for Newton's method in one dimension

The prof is so simple that it will not add much to the length of the paper: Let $F(x)=x-F(x)/F'(x)$, $x_{n+1}=F(x_n)$. By convexity, all $x_n>r$, so $F(x_n)>0$, so $x_n$ is a bounded decreasing sequence, so it has a limit. This limit must be $r$.

Numerical analysis: Mathematics of scientific computing by Kincaid and Cheney has a proof on page 86 (third edition) if you really need a reference.

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.