Global convergence for Newton's method in one dimension 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...".
 A: 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 $r$. 
A: Numerical analysis: Mathematics of scientific computing by Kincaid and Cheney has a proof on page 86 (third edition) if you really need a reference. 
A: 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.
A: $\newcommand{\+}{^{\dagger}}%
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*
  
*$\ds{x_{n + 1} = x_{n} - {\fermi\pars{x_{n}} \over \fermi'\pars{x_{n}}}.\quad}$ ( Newton-Rapson ) 
  
*Define $\ds{\epsilon_{n} \equiv x_{n} - s}$ where $\ds{s}$ is a
  root of $\ds{\fermi\pars{x} = 0}$.
  
*We consider a situation where $\ds{\verts{\epsilon_{n}}}$ is 'close' to $\ds{s}$:  
  

\begin{align}
\epsilon_{n + 1}
&=
\epsilon_{n}
-
{\fermi\pars{s + \epsilon_{n}} \over \fermi'\pars{s + \epsilon_{n}}}
\approx
\epsilon_{n}
-
{\overbrace{\fermi\pars{s}}^{\ds{=\ 0}} + \fermi'\pars{s}\epsilon_{n} + \fermi''\pars{s}\epsilon_{n}^{2}/2
 \over
 \fermi'\pars{s} + \fermi''\pars{s}\epsilon_{n}
 + \fermi'''\pars{s}\epsilon_{n}^{2}/2}
\\[3mm]&=
{\fermi''\pars{s}\epsilon_{n}^{2}/2 + \fermi'''\pars{s}\epsilon_{n}^{3}/2
 \over
 \fermi'\pars{s} + \fermi''\pars{s}\epsilon_{n}
 + \fermi'''\pars{s}\epsilon_{n}^{2}/2}
=
{\fermi''\pars{s}/2 + \fermi'''\pars{s}\epsilon_{n}/2
 \over
 \fermi'\pars{s} + \fermi''\pars{s}\epsilon_{n}
 + \fermi'''\pars{s}\epsilon_{n}^{2}/2}\,\epsilon_{n}^{2}
\end{align}
Up to second order in $\epsilon_{n}$, the last expression is reduced to:
$$
\bbox[15px,border:1px dotted navy]{\ds{\epsilon_{n + 1} \approx
\half\,{\fermi''\pars{s} \over \fermi'\pars{s}}\,\epsilon_{n}^{2}}}
$$
That is the reason we usually say that "$\tt\mbox{Newton-Rapson converges quadratically}$".
