My question is following
Let $(x_{n})$ be a positive sequence and satisfies the inequalities
$\frac{1}{K+x_{n+1}}\leq x_{n}\leq \frac{1}{e^{-v}+x_{n+1}}$
where $K>1$ and $v>0$. How to find inf $x_{n}$ and sup $x_{n}$?
My try: Iterating the given inequalities I have found upper and lower bounds for $x_{n}$ as follow
$$ \frac{\sqrt{K^{2}+4Ke^{v}}-K}{2Ke^{v}}\leq x_{n} \leq \frac{\sqrt{K^{2}+4Ke^{v}}-K}{2}. $$
But I am not so sure whether they are inf and sup?