What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$? $$\lim_{n \to\infty} \frac{n!}{n^n}$$
I have a question: is it valid to use Stirling's Formula to prove convergence of the sequence?

Let $u_n=\frac{n!}{n^n}$ then $u_{n+1}=\frac{(n+1)!}{(n+1)^{(n+1)}}$. Now, $$L=\lim_{n\to \infty}\frac{u_{n+1}}{u_n}\Rightarrow \lim_{n\to \infty}\frac{(n+1)!}{(n+1)^{(n+1)}}\times \frac{n^n}{n!}\\ =\lim_{n\to \infty}\left(\frac{n}{n+1}\right)^n=e^{-1}<1.$$ By ratio test for sequences the limit $\displaystyle \lim_{n\to \infty}\frac{n!}{n^n}=0.$
