sequence is cauchy

how to prove that the recursive sequence $a_0\ge 0$, $a_{n+1}=\frac{3(1+a_n)}{3+a_n}$ is a cauchy sequence? The sequence seems to be bounded and if the sequence is monotonic increasing (I still dont know if it is..), it is convergent, then the sequence must be cauchy. But how to prove with the definition of cauchy sequence if the sequence is cauchy? If I try to start with $|a_{n+1}-a_n|=...$ I don't do useful calculations.. Regards