The sequense $\{a_n\}$ such that $a_1>0, a_2>0$ and $$a_{n+1}=\frac2{a_n+a_{n−1}}$$ for $n≥2$.
Prove that this sequence has a limit.
I know I need to prove 1) the sequence is monotone; 2) the sequence is bounded.
But sequense is not monotone.
Any hints?
If both $a_1$ and $a_2$ are positive, then $a_n$ is also positive for $n \geq 2$. Let us suppose $\lim_{n\to\infty} a_n (\geq 0)$ exists and is unique, and call it $L$. Then: $$L = \lim_{n\to\infty} a_n = \lim_{n\to\infty} \frac{2}{a_{n-1} +a_{n-2}} = \frac{2}{\lim_{n\to\infty} a_{n-1} +\lim_{n\to\infty} a_{n-2}} = \frac{2}{L+L} = \frac{1}{L}.$$ So $L^2 =1$, or $L =\mp 1$. But we assumed $L$ to be positive.