Assume that $(a\_n)$ is a sequence of positive real numbers satisfying $a\_{n+1}=\frac{2}{a\_n+a\_{n-1}}$ for $n=2,3,\dots$.

Prove that $(a\_n)$ is convergent and find the limit.

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I have no idea how to prove convergence.
I tried to prove that $(a\_n)$ is bounded and monotone, but both trials failed.

If convergence is proven, the limit is easy: if $g=\lim a\_n$, then $g\ge 0$ because all $a\_n$'s are positive, the equivalent relation $a\_{n+1}(a\_n+a\_{n-1})=2$ gives $g(g+g)=2$, so finally $g=1$.