$a_{n+1}=1+\frac{1}{a_n}.$ Find the limit. Let $a_1=1$ and $a_{n+1}=1+\frac{1}{a_n}.$ 
Is $a_n$ convergent? 
How could i find its limit?
I found even terms of the sequence decrease and odd terms are increase. But i cant find upper and lower bounds to use monotone convergence theorem.

You can prove by induction that
$$a_n = \frac{F_{n+1}}{F_n}$$
where $\{F_n\}_{n\geq 1}=\{1,1,2,3,5,8,\ldots\}$ is the Fibonacci sequence.
Binet formula hence gives:
$$\lim_{n\to +\infty} a_n = \phi = \frac{1+\sqrt{5}}{2}.$$
