I am interested in using a result about Newton's method, which basically states that if f is convex on $[a,b]$ and it holds $f(a)<0$ and $f(b)>0$, then the Newton iteration converges to $x^*\in[a,b]$ with $f(x^*)=0$ for every starting value $x_0\in[a,b]$ with $f(x_0)\geq 0$.

The proof is not that complicated. Unfortunately, I haven't found this theorem in English literature yet (only in some German lecture notes). Could you recommend me a book or a paper, where this might be found?

Please note: I have also asked this question on math.SE (but without any success).