Proving convergence of a recursively defined sequence Using a computer it is easy to see that the sequences defined by letting $a_1=1$, $a_2=m$, and
$$a_n=\frac{2a_{n-1}a_{n-2}}{a_{n-1}+a_{n-2}}$$
converges to $\frac{3m}{m+2}$. I would very much like to know how to prove this. I don't even know where to start.

Let $A_n=\frac{a_n}{a_{n-1}}$, then we have $A_n=\frac{2}{1+A_{n-1}}$, it is easy to find that $\lim\limits_{n\to\infty }A_n$ exists 
then we have $\lim\limits_{n\to\infty }\frac{a_n}{a_{n-1}}=\frac{\sqrt{2}-1}{2}<1$, we can use the $\epsilon- N$ to illustrate the result......
