Local contraction proof for Newton iteration

For the Newton map $g(x)=x-f(x)/f^{\prime }(x)$ near a simple root $r$,Bounds $|f^{\prime }|\ge \delta$ and $|f^{\prime \prime }|\le M$ make $|g^{\prime }|<1$ on a sufficiently small closed interval around $r$. The contraction mapping theorem then gives the unique local fixed point.