Prove that the limits of these two recurring sequences is pi.

I'm trying to prove that these two sequences $a_n$ and $b_n$ converge to $\pi$, but cannot find a method of doing so. The two sequences are defined as such:

$a_0=2\sqrt3, b_0=3$

$a_n= \frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}}$

$b_n=\sqrt{a_n b_{n-1}}$

So far, I have proved that $\forall n \in \mathbb{N}, a_n>b_n$, that, if the limits of both exist, they are equal, and that $a_n$ is decreasing and $b_n$ is increasing, but I am utterly stuck at how I should prove that this limit is in fact $\pi$. I have tried playing with the definition of convergence, but still am stuck!

Any help would be greatly appreciated.