strange recurrence relation and limit

Question

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

$b_{n+1}=\sqrt{a_{n+1}b_{n}}$

also, $a_1=a$ and $b_1=b$

can someone find $$\lim_{n\to\infty}a_{n}$$ This question is from guillaume musso's novel. I remember that the answer included $\arccos$ thing..