Prove that the sequence $C_{n+1} = 1 + \frac{C_n}{C_n + 1}$ is monotonically increasing by induction I have a sequence
$$C_{n+1}  = 1 + \frac{C_n}{C_n + 1}$$
With a base case $C_1 = 3/2$ and want to prove that it's monotonically increasing by induction.
Whenever I try to prove $C_n < C_{n+1} \implies C_{n+1} < C_{n+2}$ 
where $$C_{n+2} = 1 + \frac{2C_n + 1}{3C_n + 2} $$
I get to the inequality $$ \frac{C_n}{C_n + 1} < \frac{2C_n + 1}{(C_n + 1)^2} $$
but don't know how to get to the point where that implies that $C_{n+1} < C_{n+2}$
Thank

As $C_1=\frac 32$, it is clear that every consecutive term $C_2, C_3, \dots, C_n$ is positive as $C_{n+1}$ is the sum of two positive terms for all $n>0$. We have
\begin{align}C_{n+2}-C_{n+1}&=\Big(1 + \frac{C_{n+1}}{C_{n+1} + 1}\Big) - \Big(1- \frac{C_n}{C_n + 1}\Big)\\&=
\Big(1 + \frac{1 + \frac{C_n}{C_n + 1}}{2 + \frac{C_n}{C_n + 1}}\Big) - \Big(1- \frac{C_n}{C_n + 1}\Big)\\&=
\Big(1 + \frac{2C_n+1}{3C_n+2}\Big) - \Big(1- \frac{C_n}{C_n + 1}\Big)\\&=
\frac{2C_n+1}{3C_n+2}+\frac{C_n}{C_n + 1}\\&>0
\end{align}
which shows that $C_{n+2}>C_{n+1}$.
