limits - Prove that $lim_{nto infty}{left(1+frac{1}{n^2}right)^{n^3}}=+infty$

I'm trying to prove that $\lim_{n\to \infty}{\left(1+\frac{1}{n^2}\right)^{n^3}}=+\infty$ using the fact that if $b_n\ge a_n$ eventually, and $\lim_{n\to \infty}{a_n}=+\infty$, then $\lim_{n\to \infty}{b_n}=+\infty$, where $b_n:=n^n$.

I'm struggling to show that $b_n\ge a_n$ by induction. Is this a good method? If so, what would be the best way to proceed. Thank you in advance.

Answer

Using my comment: for almost all $\;n\in\Bbb N\;$ , we have that (since $\;e=2.7...\;$)

$$2.5\le\left(1+\frac1{n^2}\right)^{n^2}\le3\implies\left[\left(1+\frac1{n^2}\right)^{n^2}\right]^n\ge(2.5)^n\xrightarrow[n\to\infty]{}\infty$$