Proof by induction question - inductive step clarification?

Hi,

I'm doing some revision, and am asked to prove by induction:

$\displaystyle n^n > 2^{n+1} \text{ for all } n \geqslant 3$

The provided solution has the obvious base step (n = 3), the following hypothesis:

$\displaystyle \text{Let } n \geqslant 3 \text{ such that } n^n > 2^{n+1}$

and the following step:

$\displaystyle (n + 1)^{n+1} > n^n \cdot n >_{IH} 2^{n+1} \cdot n > 2^{n+2}$

Regrettably I can't see how the inductive step leads to prove the hypothesis, and so am wondering if someone would be kind enough to break it down for me?

Thanks in advance,

James