# Two sequences defined for all naturals by

1. Jan 13, 2005

### quasar987

Consider $x_1,y_1 \in \mathbb{R}$ such that $x_1>y_1>0$ and $\{x_n\},\{y_n\}$ the two sequences defined for all naturals by

$$x_{n+1}=\frac{x_n+y_n}{2}, \ \ \ \ \ y_{n+1}=\sqrt{x_n y_n}$$

Show that the sequence $\{y_n\}$ is increasing and as $x_1$ for an upper bound.

I would appreciate some help on this one, I have made no progress.

2. Jan 13, 2005

### mathman

The key to the proof is to show xn+1>yn+1. Since they are both positive, this is equivalent to showing the squares exhibit the same inequality. A simple calculation will show this to be true.

xn+12-yn+12=((xn-yn)/2)2.

The rest is easy. xn>xn+1> ,yn<yn+1.