# Sequence generated by arithmetic and harmonic mean converges to square root

I need to show the following:

Starting with a point $S=(a,b)$ of the plane with $a>b>0$, we generate a sequence of points $(x_n,y_n)$ according to the rule: $$x_0=a, y_0=b, x_{n+1}=\frac{x_n+y_n}{2}, y_{n+1}=\frac{2x_ny_n}{x_n+y_n}.$$ Prove that $\lim x_n=\lim y_n=\sqrt{ab}$.

I can show that $x_ny_n=ab$ and $x_n>y_n$ for all $n$, but I am having trouble proving $\lim x_n=\lim y_n=\sqrt{ab}$. Suggestions?

• First prove convergence. Then use the recurrence to find the limit. – J.R. Feb 16 '14 at 18:29

## 1 Answer

Let us check that $(x_n)_n$ is a convergent sequence. It is monotonously decreasing:

$$x_{n+1}=\frac{x_n+y_n}{2}<\frac{x_n+x_n}{2}=x_n$$

On the other hand the sequence is bounded from below: $x_{n}\ge 0$.

Therefore it converges to some limit $x=\lim_{n} x_n$ (monotone convergence theorem). Since $x_ny_n=ab$ for all $n$, also $(y_n)_n$ converges to some limit $y=\lim_n y_n$. Now the recurrence for $x_n$ implies

$$x=\frac{x+y}{2}$$

That is, $x=y$. Since $xy=ab$, we have $x=y=\sqrt{ab}$.