Find the limit of a sequence defined recursively as $x_1=\frac{3}{2}$, $x_{n+1}=\dfrac{4+3x_n}{3+2x_n}$ with $n\in \mathbb{N}$. Show that the limit exists before attempting to find it.
I need to show that it’s bounded below by ${\sqrt 2}$ , which I failed , but assuming that it is I was able to also show its decreasing so hence the existence of a limit. Any help on showing it’s bounded below by ${\sqrt 2}$ would be greatly appreciated . Thanks.
Best Answer
Expanding $x_{n+1}^2-2$ yields $$x_{n+1}^2-2=\frac{x_n^2-2}{(3+2x_n)^2}$$ Noting that $x_1 > \sqrt{2}$, it follows by induction on $n$ that $x_n > \sqrt{2}$ for all positive integers $n$.