 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2014/discrete-mathematics-continuous-mathematics-and-probability/5/b/ii/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

x_{n} \geq a for every n \geq 1.

This holds even when

0 < x_{0} < a

: the first update moves to or above the positive square root.

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