Solve by induction: $n!>(n/e)^n$

Here's a hint. You assume that $n!>\left(\frac{n}{e}\right)^n$. Now you should show it for n+1, i.e., you should show that $(n+1)! > \left(\frac{n+1}{e}\right)^{n+1}$.

You can write

\begin{equation} (n+1)! > (n+1) \left(\frac{n}{e}\right)^n = (n+1)\left(\frac{n}{n+1}\right)^n \left(\frac{(n+1)^n}{e^n}\right) \end{equation}

Can you solve it from here?

By considering the exponential power series we observe that for $x>0$, $$ e^x > \frac{x^n}{n!} $$ Now setting $x=n$ we obtain $$e^n > \frac{n^n}{n!} $$ which rearranges to precisely what is desired. I should note that I had first learned this incredibly short and simple proof of this fact from Qiaochu Yuan's posts on this website, and he in turn attributed it to this article written by Terence Tao.