Calculate big O notation n!

Question

How to prove that $n!=O(5^n)$ is false?

I tried to use mathematical induction, but I wasn't able to find any results.

Answer 1

Just observe that, using $a_n=\frac{n!}{5^n}$,

$$\lim_{n\to \infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to \infty}\frac{(n+1)!5^{n}}{n!5^{n+1}}=\lim_{n\to \infty}\frac{n+1}{5}=\infty,$$ thus $\lim_{n\to \infty}a_n=\infty$ so that we cannot possibly have $\frac{n!}{5^n}\leq C$ for a $C\in\mathbb{R}$ as $n\to\infty$.