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Show that |x_n - √2| < 1/2^n for n>2

Source: Canada National Mathematical Olympiad 1985 - Problem 5

September 30, 2011
inductioninequalitiestriangle inequalityinequalities proposed

Problem Statement

Let 1<x1<21 < x_1 < 2 and, for n=1n = 1, 2, …\dots, define xn+1=1+xn−12xn2x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2. Prove that, for n≥3n \ge 3, ∣xn−2∣<2−n|x_n - \sqrt{2}| < 2^{-n}.
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