CMO 1985 Problem 5

Let $1 < x_1 < 2$ and, for $n = 1$ , 2, $\dots$ , define $x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2$ . Prove that, for $n \ge 3$ , $|x_n - \sqrt{2}| < 2^{-n}$ .