Consider the following system of two equations in two variables,

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

where

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

Show that if x^0 is any vector which satisfies

r_1(x) = 0 and x_1 < x_2,

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