Showing $1$ is a limit of a sequence

Question

Let $x_n$ be a sequence such that:
$x_1 = {3 \over 2}$ and $x_n = {3 \over {4-x_{n-1}}}$.

I already showed by induction that $x_n$ is strictly decreasing and bounded below by $1$. Is is suffice in order to prove $1$ is the limit of $x_n$? Should I also show that $1$ is the greatest lower bound?

Answer 1

It's not suffice to prove that 1 is the limit of $x_n$. but you know that $x_n$ is strictly decreasing and that it is bounded. this implies that the limit of this sequence exists. Hint: Further should be $\lim_{n \to \infty} x_n = \lim_{n \to \infty} x_{n+1}$.

Answer 2

By the monotone convergence theorem, if a sequence is decreasing and bounded below by an infimum, the sequence will converge to the infimum.