Showing the sequence converges $a_{1}=\frac{1}{2}$, $a_{n+1}=\frac{1}{2+a_{n}}$ Showing the sequence converges $a_{1}=\frac{1}{2}$,  $a_{n+1}=\frac{1}{2+a_{n}}$.
I already know that if $(a_{n})$ converges then it does to  $\sqrt{2}-1$.But i dont't know how to prove that this sequence cenverges.
EDIT
I think that the subsequence $(a_{2n+1})$ is monotonic decreasing and the subsequence $(a_{2n})$ is monotonic increasing.

Since you already have a reasonable conjecture consider the sequence $(b_n)_{n\geq1}$ defined by
$$a_n=\sqrt{2}-1+b_n,\qquad{\rm resp.,}\qquad b_n=a_n+1-\sqrt{2}\qquad(n\geq1)\ ,$$
and try to prove that $\lim_{n\to\infty} b_n=0$, which should be simpler.
