How to prove divergent sequences? For this question, I know that the sequence diverges to infinity, but I'm not sure if I am doing it right. Here is what I have so far. Can anyone please help me out?
Determine whether the following sequence is convergent or divergent
$a_n = \{8n^3 + n^2 -2\}$
$\lim_{n \to \infty} a_n = \infty$
Wts for any $M>0$, there exists some $N>0$, st if $n>N$, then $a_n>M$.
$n^3 > N^3 > M$
$n^3(8 + \frac{1}{n} - \frac{2}{n^3}) > N^3(8 + \frac{1}{N} - \frac{2}{N^3}) > M$
$n^3(8 + \frac{1}{n} - \frac{2}{n^3}) > N > (\frac{M}{8+\frac{1}{N} -\frac{2}{N^3}})^\frac{1}{3}$

We fix $M>2$ and we look for $N$ such that $\forall n>N$
$$a_n = 8n^3 + n^2 -2>M$$
then choose $n=M$ and check that
$$8M^3 + M^2 -2>M\iff 8M^3 + M^2>M+2$$
which is true.
Then it suffice to choose $N\ge M$.
