proving convergence of $a_{n+1}=1+\frac{1}{1+a_{n}}$ 
$a_1=1.$ $a_{n+1}=1+\frac{1}{1+a_{n}}$

Prove that the sequence is convergent.
I'm trying to prove the convergence of this sequence but having trouble. At first I thought this might be a monotone sequence since then I can try monotone convergence theorem to prove its convergence.
But after checking some terms, I realized it seemed the sequence is oscillating. So I'm not sure how to prove the convergence of this sequence.
Thanks.

A method often useful with oscillating sequences: Let $b_n=|(a_n)^2-2|.$ Then $$0\le b_{n+1}=\frac {b_n}{(1+a_n)^2}\le \frac {b_n}{4} $$ because $1+a_n\ge 2$ by induction on $n$.
So $b_n$ decreases to  $0$. So $(a_n)^2\to 2$ with each $a_n>0.$
The motivation for the "$2$" in the definition of $b_n$ is that IF $a_n$ converges to a limit $L$ then $L=\lim_{n\to \infty}a_{n+1}=\lim_{n\to \infty} 1+\frac {1}{1+a_n}=1+\frac {1}{1+L},$ implying $L^2=2.$
