How should I prove that $c^n/n!$, where c is a constant, tends to zero for a large n.

Question

I want to prove that $$\frac{c^n}{n!}$$ tends to zero when $n$, a positive integer, is quite large. I do not think I can use L'Hospital's rule here as $n$ is discrete. Should we assume $n$ to be continuous to sove this problem. Furthermore, I do not find approximating $n!$ with gamma function of much help.

Answer 1

Suppose $c=2$ for convenience. Then $$ \frac{2^n}{n!}=\frac{2}{n}\cdot\frac{2}{n-1}\cdot\ldots\cdot\frac{2}{3}\left(\frac{2}{2}\cdot\frac{2}{1}\right)\leq\left(\frac{2}{3}\right)^{n-2}\left(\frac{2}{2}\cdot\frac{2}{1}\right)\xrightarrow{n\to\infty}0. $$ This can easily be generalized to any $c>0$.