Exercise 16, § 5

Let $x_0, y_0$ be two real numbers such that $0 < y_0 \leq x_0$. For $n > 0$ we define two sequences $(x_n), (y_n)$ inductively by the relations

Show that the two sequences $(x_n), (y_n)$ tend to the same limit $\alpha$ (the "arithmetico-geometric mean" of $x_0, y_0$); also there exist numbers $a > 0$ and $\gamma$ such that $0 < \gamma < 1$ and $x_n - y_n \leq a \gamma^{2^n}$ for each $n$ [observe that $x_{n+1} - y_{n+1} = (x_n - y_n)^2 / 4(x_{n+1} + y_{n+1})$].