Problem 5 Paper Let 1<x1<21 < x_1 < 21<x1<2 and, for n=1n = 1n=1 , 2, …\dots… , define xn+1=1+xn−12xn2x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2xn+1=1+xn−21xn2 . Prove that, for n≥3n \ge 3n≥3 , ∣xn−2∣<2−n|x_n - \sqrt{2}| < 2^{-n}∣xn−2∣<2−n . Answer, solutions and source