$$U(n+1) = (6+U(n))^{1/3},\text{ and } U(0) = 1.$$
- Prove by induction that for all positive integers $n, U(n)$ is increasing.
- Prove by induction that for all positive integers $n, U(n) \leq 2$ (namely, the sequence is bounded from below).
- Does the sequence $U(n)$ have limit? If yes, what is it?
So I did $U(1) = (6+1)^{1/3} = (7)^{1/3}$ which checks off but where can I go from there?
Hints:
Start with proving by induction that $0\leq U\left(n\right)<2$ for each $n$.
$0\leq x<2$ implies that $\left(6+x\right)^{\frac{1}{3}}>x$.
It there is indeed a limit $z$ then $\left(6+z\right)^{\frac{1}{3}}=z$