Show that $\inf \{\frac{2^n}{n!}:n\in\mathbb{N}\}= 0$

Question

Show that $$\inf \left\{\frac{2^n}{n!}:n\in\mathbb{N} \right\}= 0$$

My try:

The following inequality is true:

$0 \leq \frac{2^n}{n!} \leq \frac{10}{n}$

and I also know that the inf$\{\frac{10}{n}\} = 0$

then, useing the fact that if $0 < a_n < b_n\quad$ then if $\quad$ inf$\{b_n\} = 0\quad \Rightarrow\quad $ inf$\{a_n\} = 0$,

inf$\{\frac{2^n}{n!}:n\in\mathbb{N}\}= 0$

Answer 1

Your solution is correct, but it needs more justification at this step:

$$0 \leq \frac{2^n}{n!} \le \frac{10}{n}$$

How would you prove this?