Prove $u_{n}$ is decreasing $$u_{1}=2, \quad u_{n+1}=\frac{1}{3-u_n}$$
Prove it is decreasing and convergent and calculate its limit.
Is it possible to define $u_{n}$ in terms of $n$?
In order to prove it is decreasing, I calculated some terms but I would like to know how to do it in a more "elaborated" way.

If there is a limit, it will be defined by $$L=\frac{1}{3-L}$$ which reduces to $L^2-3L+1=0$. You need to solve this quadratic and discard any root greater than $2$ since this is the starting value and that you proved that the terms are decreasing.
