Solution
= Solution
The update in <Newton's method> is exactly $x_n=g(x_{n-1})$. Positivity is preserved, and the <arithmetic-geometric mean inequality> from part (a) applies to every update. Consequently
$$
\boxed{x_n\geq\sqrt a\quad\text{for every }n\geq1.}
$$
This holds even when $0<x_0<\sqrt a$: the first update moves to or above the positive <square root>.