Prove that sequence $a_{n+1} = \frac{1}{1+a_n}$ is bounded? Consider sequence; 
$$a_1 = 1$$
$$a_{n+1} = \frac{1}{1+a_n}$$
Prove that sequence is bounded.
It is a question from my test. I couldn't figure out how to solve it.

If $a_n$ is positive then
$$a_{n+1} = \frac{1}{1 + a_n} > 0.$$
Also, if $a_n$ is positive, then
$$1 < 1 + a_n \implies 1 > \frac{1}{1 + a_n} \implies 1 > a_{n+1}.$$
Therefore, if $a_{n} > 0$ then
$$0 < a_{n + 1} < 1.$$
That is, if all the elements of the sequence are positive then the sequence is bounded. Since the first element is positive ($a_1 = 1$), the first equation proves by induction that all the elements will be positive.
