I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$.
My steps are:
- induction base case: $n=0:$ $0² \leq 2⁰$ which is okay.
- inductive step: $n \rightarrow n+1:$ $(n+1)²\leq2^{n+1}$ $$(n+1)^2 = n^2 + 2n + 1 = ...help...\leq 2^{n+1}$$
I know the bernoulli inequality but don't know where to use it, if I even need to. I have problems when it comes to proving things which are based on orders..