Question: Prove that $n \leq (\frac{n^3}{n^2+2})^c$ for all non zero positive integer $n \geq n_0$ and some positive non zero constant $c$. (This is relating to the Big O proof)
Attempt: let $c=2$ and $n_0 = 1$, then for all $n \geq 1$:
$n \leq (\frac{n^3}{n^2+2})^2 \rightarrow n^5+4n^3+4n \leq n^6 $
I don't really know how to prove from here on, maybe I should use induction. But I'll appreciate it very much if anyone can give me a hint and point me to the right direction.
Best Answer
$n^{5}+4n^{3}+4n \leq n^{5}+4n^{5}+4n^{5}=9n^{5} \leq n^{6}$ for $ \geq 9$. So take $n_0=9$ instead of $n_0=1$.