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.

Take $\log$ from both sides $a \log n < n \log a $
Now lets check who  grows faster $\lim\limits_{n\to \infty}\frac{n \log a}{a \log n}=\infty$ and $\lim\limits_{n\to \infty}\frac{a \log n}{n \log a}=0$ 
