MathNet 1985 — Part 01 Question 9

Let $1 \lt x_{1} \lt 2$ and, for $n = 1, 2, \ldots$, define $x_{n+1} = 1 + x_{n} - \frac{1}{2} x_{n}^{2}$. Prove that, for $n \geq 3$, $\left| x_{n} - \sqrt{2} \right| \lt 2^{-n}$.