Bounding and Sequence Convergence Proofs

Question

I am trying to prove that the sequence $\{\frac{1-n^2+n^3}{n^3-1}\}$ converges to 1. But I am having trouble finding a sequence that can bound this one. So far, I have done this,

$|\frac{1-n^2+n^3}{n^3-1}-1|=|\frac{1-n^2+n^3}{n^3-1}-\frac{n^3-1}{n^3-1}|= |\frac{n^2}{n^3-1}|$

as an attempt to find a way to write $\epsilon$ in terms of $N$.

Any help would be greatly appreciated.

Answer 1

Now, since for $n\geq2$ we have $n^3-1>\frac{n^3}{2}$, we obtain: $$\left|\frac{1-n^2+n^3}{n^3-1}-1\right|=\left|\frac{n^2-2}{n^3-1}\right|<\frac{2}{n}.$$ can you end it now?