convergence of sequence defined recursively $\frac{2}{a_{n+2}}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$ How to prove that the sequence ${a_n}$ defined by $\frac{2}{a_{n+2}}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$ for $n\geq1$ and $0\lt{a_1}\lt{a_2}$ converges? How to find its limit?  I truly do not know how to proceed further. I tried to use A.M>G.M inequality but with no success. 

Note that $2\left(\frac{1}{a_{n+2}}-\frac{1}{a_{n+1}} \right)=-\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n}}\right)$, thus with $b_n = \frac{1}{a_{n+1}}-\frac{1}{a_{n}}$, $b_{n+1}=-\frac 12 b_n$.
Hence $b_{n+1} = \frac{(-1)^n}{2^n}b_1$, so $\sum_n b_n$ converges, thus $\frac1{a_n}$ converges, and so does $a_n$.
Let $\ell=\lim_n a_n$. Then  $\sum_{n=1}^\infty b_n = \frac 1\ell -\frac 1{a_1}$, which rewrites as $\frac 23 \left(\frac 1{a_1} - \frac 1{a_2} \right) = \frac 1\ell -\frac 1{a_1}$, hence $$\ell = \frac{1}{\frac 23 \left(\frac 1{a_1} - \frac 1{a_2} \right)+\frac 1{a_1}}$$
