So I have a question. This isn't a particularly hard proof.

I had previously proved that: $\displaystyle n < 2^n$

So i'm allowed to use that. Not allowed to use derivatives/graphs either.

$\displaystyle n^3 \le 3^n \forall n \in \mathbb{Z}_{>0}$

Now the question is can anyone prove this more efficiently than triple induction.

my first induction statement obviously is $\displaystyle n^3 \le 3^n$ I base cased out 1, and 2, then I assumed it was true for all n > 2.

the second induction statement became $\displaystyle 3n^2+3n + 1 \le 2*3^n$ for n > 2.

the third induction statement became $\displaystyle 3n+3 \le 2*3^n$ and since I n >2 for the above 2, n > 2 for this as well.

I was wondering if anyone can do this faster than 3 inductions.