Prove that a given recursion sequence converges
We prove by induction that:
- $1<x_n<3$
- $x_n$ is decreasing.
The base case is obvious. Now assume that $1<x_{n-1}<3$ for some $n$. Then $$ \frac{3}{4-1}< \frac{3}{4-x_{n-1}}<\frac{3}{4-3} $$ or, after simplifying, $1<x_n<3$, so $1.$ holds for $n$. Also, note that $1<x_{n-1}<3$ implies $$ (x_{n-1}-1)(x_{n-1}-3)<0\Rightarrow 3<4x_{n-1}-x_{n-1}^2 $$ so $$ x_n=\frac{3}{4-x_{n-1}}<x_{n-1} $$ So $2.$ holds as well. Now by the monotone convergence theorem, $x_n$ converges. With a little more work, we can show that this limit is actually $1$.