Let $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuously differentiable, $F(x')=0$ and let $x^{(0)}$ be so that
a) The set $K:=\{x :||F(x) \leq ||F(x^{(0)})||\}$ is compact.
b) For every $x \in K \backslash \{x'\}$ there exists $F'(x)^{-1}$ and
$\lim \sup_{x \rightarrow x'} ||F'(x)^{-1}||||x-x'|| < \infty$
Show that the damped Newton's method with initial value $x^{(0)}$ converges to a root of F.
The only thing I know is that $F(x^{(k)})$ converges to 0 if the vector of Newton directions is limited. But how could this help me and how do I make a statements about the convergence of $x^{(k)}$.