calculus - Prove $a_1 = 1, a_{n+1} = frac{1+a_n}{2+a_n}$ converges

I need to prove that the sequence defined by

$$a_1 = 1, a_{n+1} = \frac{1+a_n}{2+a_n}$$
converges.

I tried to prove that it's bounded and monotonically decreasing, but I couldn't prove it's monotonically decreasing.
I also managed to find the limit assuming it converges.

Answer

Note that
$$a_{n+1} = \frac{1+a_n}{2+a_n} = 1 - \frac{1}{2 + a_{n}}.$$
So, given that both $a_n$ and $a_{n-1}$ are positive, we have
$$
a_{n} < a_{n-1} \implies \frac {1}{2 + a_n} > \frac 1{2 + a_{n-1}}

\implies 1 - \frac {1}{2 + a_n} < 1 - \frac 1{2 + a_{n-1}} \implies
a_{n+1} < a_n.
$$
So, we can indeed conclude that the sequence is monotonic.