This is driving me crazy, because it seems wrong but I haven't found it in any of the errata.

Problem:

Suppose $\displaystyle f$ is twice differentiable on $\displaystyle [a, b], f(a) < 0, f(b) > 0, f'(x) \ge \delta > $0, and $\displaystyle 0 \le f''(x) \le M$ for all $\displaystyle x \in [a, b]$. Let $\displaystyle \xi$ be the unique point in $\displaystyle (a, b)$ at which $\displaystyle f(\xi) = 0$.

Define

$\displaystyle x_{n + 1} = x_n - \frac{f(x_n)}{f'(x_n)}$.

Prove that $\displaystyle x_{n + 1} < x_n$, and that

$\displaystyle \lim x_n = \xi$.

EDIT: Assume $\displaystyle x_0 \in (\xi, b)$

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My issue is that I can only show $\displaystyle x_n \ge x_{n + 1} \ge \xi$ i.e. I can't get a strict inequality. I would think there is a counterexample if we take $\displaystyle f(x) = x$ so that regardless of which $\displaystyle x_1$ you choose, $\displaystyle x_2 = x_3 = ... = 0$.