# Mathematical Induction ($2^n\ge n^4$ for $n\ge n_0$)

I'm trying to prove a statement with M.I. Here is my statement:

There exist an $n_{0}\epsilon\mathbb{N}$ such that $2^{n}\geq n^{4}$ for all $n\geq n_{0}$

Well, when I start to prove:

Initial step: Let n = $n_{0}$

I get $2^{n_0}\geq n_{0}^{4}$ . But I realize this equation is just true when $n_{0}$ equal to $0$, $1$.

So I mean I can't start because I get a problem at the initial step. Any advice? How can I solve this so I can continue to prove.

First, you have to work out what $n_0$ is. This you do by trying various values of $n$ until you are convinced that you have found a suitable $n_0$. Also, there seems to be some confusion as to whether you are trying to show $2^n\ge n^2$ or $2^n\ge n^4$. – Gerry Myerson Dec 17 '12 at 4:50
Your inequality holds for $n \geq 16$. Hence, choose your $n_0 = 16$.