Consider the following system of two equations in two variables,

$\displaystyle r(x) = 0$, with $\displaystyle r(x) := (r_1(x) \ r_2(x))^T$, $\displaystyle x := (x_1 \ x_2)^T$,

where

$\displaystyle r_1(x) := x_1 + x_2 - 1 = 0$
$\displaystyle r_2(x) := (2x_1 - 1)^2 + (2x_2 - 1)^2 - \frac{2}{3}$.

Show that if $\displaystyle x^0$ is any vector which satisfies

$\displaystyle r_1(x) = 0$ and $\displaystyle x_1 < x_2$,

then the sequence of iterates {$\displaystyle x^k$} generated by Newton's method applied to $\displaystyle r(x) = 0$ converges.