Let $0<y_1<x_1 $ and set $x_{n+1} = \frac{x_n+y_n}{2}$ and $y_{n+1}=\sqrt{x_ny_n}$, $n\in\Bbb{N}$

Question

Let $0<y_1<x_1 $ and set $x_{n+1} = \frac{x_n+y_n}{2}$ and $y_{n+1}=\sqrt{x_ny_n}$, $n\in\Bbb{N}$.

1. Prove that $0<y_n<x_n$ $\forall n\in\Bbb{N}$.
2. Prove that $y_n$ is increasing and bounded above, and $x_n$ is decreasing and bounded below.
3. Prove that $0<x_{n+1}-y_{n+1}<\frac{(x_1-y_1)}{2^n} $ for $n\in\Bbb{N}$.
4. Prove that $\lim_{x\to \infty}{x_n}=\lim_{y\to \infty}{y_n}$.

For the first one I tried to prove by induction but I found $x_n<y_n$.

For the second one:
$y_{n+1}=\sqrt{x_ny_n}\implies y_{n+2}=\sqrt{x_{n+1}y_{n+1}}=\sqrt{\frac{x_n+y_n}{2}.\sqrt{x_ny_n}}$
So, $y_{n+2}^2>y_{n+1}^2$ which implies $y_n$ is increasing??? I did same thing for $x_n$.

Answer 1

1. By AM > GM, $x_n$ > $y_n$. Or, $$4x_{n+1}^2-4y_{n+1}^2 = (x_n-y_n)^2 > 0$$

2-1. $$x_{n+1} > y_{n+1}$$ $$\frac{x_n+y_n}{2} > \sqrt{x_ny_n}$$ $$\frac{x_n+y_n}{2}\sqrt{x_ny_n} > {x_ny_n}$$ $$\sqrt{\frac{x_n+y_n}{2}\sqrt{x_ny_n}} > \sqrt{x_ny_n}$$ $$\sqrt{x_{n+1}y_{n+1}} > \sqrt{x_ny_n}$$ $$\therefore y_{n+2}>y_{n+1}$$

2-2. $$x_{n+1}=\frac{x_{n}+y_{n}}{2}<\frac{x_{n}+x_{n}}{2}=x_{n}$$

$y_n$ increases and $x_n$ decreases, but $y_n<x_n$. Therefore $y_n$ is upper bounded and $x_n$ is lower bounded.

3. $$x_{n+1}-y_{n+1}=\frac{x_{n}+y_{n}}{2}-\sqrt{x_ny_n}=\frac{x_{n}-2\sqrt{x_ny_n}+y_{n}}{2}<\frac{x_{n}-2\sqrt{y_ny_n}+y_{n}}{2}=\frac{x_n-y_n}{2}$$

$$x_{n+1}-y_{n+1}<\frac{x_n-y_n}{2}<\frac{x_{n-1}-y_{n-1}}{2^2}<\cdots<\frac{x_1-y_1}{2^n}$$

4. $$0<x_{n+1}-y_{n+1}<\cdots<\frac{x_1-y_1}{2^n}$$ $$\lim_{n\to\infty}0\le \lim_{n\to\infty}x_{n+1}-y_{n+1}\le \lim_{n\to\infty}\frac{x_1-y_1}{2^n}=0$$ $$\therefore \lim_{n\to\infty}x_{n+1}-y_{n+1} = 0$$

Answer 2

You can prove $y_n<x_n$, $(x_n)$ is decreasing and $(y_n)$ is increasing in one inductive step:

• $x_{n+1}=\dfrac{x_n+y_n}2<\dfrac{2x_n}2=x_n$ by the inductive hypothesis.
• $y_{n+1}=\sqrt{x_ny_n}>\sqrt{y_n y_n}=y_n$ by the same inductive hypothesis.
• $y_{n+1}=\sqrt{x_ny_n}\le\frac{x_n+y_n}2=x_{n+1}$ by the A.G.M. inequality. The inequality is strict by the case of equality in the A.G.M. inequality: it happens only in case $x_n=y_n$, which is false by the inductive hypothesis.