The Newton–Raphson method for finding roots of a differentiable function $f : \mathbb{R} \to \mathbb{R}$ is to iterate the function $x \mapsto g(x):=x - \frac{f(x)}{f'(x)}$ Where the following conditions is assumed on $f$
- $f$ has a continous second derivative
- $f'(x) \neq 0 \ \forall \ x \in \mathbb{R}$
- There exists some $\alpha \in (0,1)$ such that $\left| f(x) f''(x)\right| \, \leq \, \alpha \left| f'(x) \right|^2 \ \forall \ x \in \mathbb{R}$
The question is how to prove that $g$ is a contraction.
I have already tried to use the definition of a contraction, that there exists some $0<\alpha<1$ such that
$d(Tx,Ty) \leq \alpha d(x,y),$
holds. Now I tried using the metric induced by the supremum norm. But alas this gave me nothing. The hint was to use the Mean Value Theorem, but I can not quite see how that applies here. Can someone give me some clear suggestions on how to proceed?