Proof of inequalities (relating to Big O notation)

Question

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.

Answer 1

$n^{5}+4n^{3}+4n \leq n^{5}+4n^{5}+4n^{5}=9n^{5} \leq n^{6}$ for $ \geq 9$. So take $n_0=9$ instead of $n_0=1$.

Answer 2

Let $n_0=4$ and $c=2.$ If $n\ge 4$ then $ n^2+2<2n^2$ and hence $$ \left(\frac {n^3}{n^2+2}\right)^2>\left(\frac {n^3}{2n^2}\right)^2=\frac {n^2}{4}=n\cdot\frac {n}{4}\ge n.$$