Proof problem: show that $n^a < a^n$ for all sufficiently large n I would like to show that $n^a < a^n$ for all sufficiently large $n$, where $a$ is a finite constant.  
This is clearly true by intuition/graphing, but I am looking for a rigorous proof.  Can anyone help me out? Thanks.

It is enough to prove $\log(n^a)=a\log n<\log(a^n)n\log a$ for all sufficiently large $n$, i.e.
$$\frac{\log n}n<\frac{\log a}a,$$
which is true for all  sufficiently large $n$ since $\dfrac{\log n}n\to 0$ as $n\to\infty$.
