Finding a limit of two given series

I got two series defined as following:

$\displaystyle u(n+1) = \dfrac{u(n) + v(n)}{2} \text{with u(0)=a} $

$\displaystyle \dfrac{1}{v(n+1)} = \dfrac{1}{2}(\dfrac{1}{u(n)} + \dfrac{1}{v(n)}) \text{with v(0)=b} $

a and b are positive real numbers.

These two series converge to the same limit. Which I'm supposed to find. I managed to prove that these two series are convergent and even adjacent. But I don't see how to get the limit. All I found out by inserting numbers was that the limit probably is $\displaystyle \sqrt{ab} $