I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge problem or that this problem is of some significance. The sequences, I believe, are suppose to equal $\pi$, so that is suggesting to me it's more of a significance.
I tried searching for the problem, as I figured anything that equals $\pi$ is no coincidence and must be a famous example. But, 'Archimedes Pi Sequence' and other variants yield much different results than this problem. I was hoping someone could either provide a link to the proof (assuming it is indeed famous) or provide a relative outline so I could try to follow along/work through it.
The problem is:
[Archimedes] Suppose that $x_0=2\sqrt{3}, \quad y_0=3$, $$x_n=\frac{2x_{n-1}y_{n-1}}{x_{n-1}+y_{n-1}},$$ and $$y_n=\sqrt{x_{n}\cdot y_{n-1}}$$ for $n\in \mathbb N$.
Prove that $x_n \downarrow x$ and $y_n \uparrow y$, as $n\to \infty$, for some $x,y \in \mathbb R$.
Next, prove $x=y$ and that $3.14155\lt x\lt 3.14161$. (This is why I believe $x=y=\pi$)
Edit: I understand that if we can show $x_n$ is decreasing and converging to $x$, and $y_n$ is increasing and converging to $y$, we can recognize the Monotone Convergence Theorem. After which, apply the limit to both equations to get $x$ and $y$. The problem I am having is understanding how we get there, and after we are there, how we show it equals $\pi$. Though, if I understand how we get there the $\pi$ part might come together.
The limit is in fact $\pi$.
Hint:
Let $a_n$ and $b_n$ be the half the perimeter of the of the inscribed and circumscribed $n$-gons of the unit circle. Then
$$b_{2n}=\frac{2a_nb_n}{a_n+b_n},$$
$$a_{2n}=\sqrt{b_{n+1}a_n}.$$
Further, $b_3=2\sqrt3$.
Historical note: Gauss used a similar process to compute (approximate) values of elliptic integrals.