I know that $\displaystyle \lim \limits_{n \to \infty } \frac{n!}{{n^n }} = 0$

since $\displaystyle \lim \limits_{n \to \infty } \frac{n!}{{n^n }} =

\lim_{n \to \infty} \left( \frac{1}{n} \cdot \frac{2}{n} \cdot \ \cdots \ \cdot \frac{n-1}{n} \cdot \frac{n}{n}\right)$ and each limit of this product goes to 0. However, how would you write an $\displaystyle \varepsilon$-proof for this limit?