This is part of a larger problem:
Prove that the sequence defined by $x_1=3$, $x_{n+1}={1\over{4-x_n}}$ converges.
I want to show that it is bounded below (by $0$ or something) and that it is decreasing, so by the Monotone Convergence Theorem it converges.
Does this seem like a valid strategy? My main concern is that I am not sure how to show the sequence is decreasing. I was thinking induction, but I don't know how to set that up for this case.