The question is as follows:
Let $a_{0} = 2\sqrt{3}$ and $b_{0} = 3$ and define two sequences recursivly as $$a_{n} = \frac{2a_{n - 1}\cdot b_{n - 1}}{a_{n - 1} + b_{n - 1}} \text{ and }b_{n} = \sqrt{a_{n}b_{n - 1}}$$
Prove that $(a_{n})$ is monotonically decreasing and is convergent. and prove that $(b_{n})$ is monotonically increasing and convergent. Then prove that they converge to $\pi$.
I managed to prove that the sequences are convergent and monotonically increasing/decreasing but I do not know how to prove they converge to $\pi$. Further, I do not have a rigorous definition for $\pi$, I only know that it is the area of a circle of radious $1$. The question also mentions as a hint that it is somehow related to Archimedes.