Newton's method for a vector field

Hint.

The process

$$ x_k = x_{k-1}-Df^{-1}(x_{k-1})f(x_{k-1}) $$

can be written as

$$ x_k = \phi(x_{k-1})\\ x_{k-1} = \phi(x_{k-2}) $$

and then

$$ |x_k-x_{k-1}| = |\phi(x_{k-1})-\phi(x_{k-2})| = |\phi'(\xi)||x_{k-1}-x_{k-2}| $$

for a suitable $\xi\in (x_{k-1}-x_{k-2})$ hence is sufficient that $|\phi'(\xi)| < 1$ for convergence