We define the Arithmetic-Harmonic mean of $a,b \in \mathbb{R_+}$ such that \begin{gather*} a_{n+1} = \frac{1}{2}(a_n + b_n) \\ b_{n+1} = \frac{2a_{n}b_{n}}{a_{n} + b_{n}} \end{gather*}
Let us also assume that $a \neq b$. I am trying to prove that the limit of these sequences exist, they are equal to each other, and as $n \rightarrow \infty$, the limit is equal to $\sqrt{ab}$, the Geometric Mean of a,b.
This is stated by mathworld here: http://mathworld.wolfram.com/Arithmetic-HarmonicMean.html
Since I have proven that AM $\geq$ HM, with equality only holding at a = b, we can see that $a_1 \geq a_{2} \geq ... \geq a_n$. In the same vein we have $b_1 \leq b_{2} \leq ... \leq b_n$.
How can I rigourously show that these are monotonic sequences bounded above and below resepctively and therefore converge to each other?
Secondly, can anyone give me a hint as to how I can show that this sequence converges to the Geometric Mean?