<s>[INST] How to prove $\lim_{n\rightarrow \infty} {a^n \over n!}=0$ 
Possible Duplicate:
How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$ 

As the topics, how to prove $\lim\limits_{n\rightarrow \infty} \dfrac{a^n}{n!}=0$
$\forall a  \in \mathbb{R^+}$
 [/INST] Let's suppose $a$ is... I don't know, $5$ for a moment. And let's look at the sequence.
$\dfrac{5}{1}, \dfrac{5^2}{2\cdot 1}, \dfrac{5^3}{3\cdot 2}, \dfrac{5^4}{4 \cdot 3 \cdot 2}, \dfrac{(5) \cdot 5^4}{(5) \cdot 4 \cdot 3 \cdot 2}, \dfrac{(5^2) \cdot 5^4}{(6 \cdot 5) \cdot 4!}, \dfrac{(5^3) \cdot 5^4}{(7 \cdot 6 \cdot 5) 4!}, \ldots$
So in particular, after $n = 5$, we have a constant multiplied by something bounded by $\left(\frac{5}{6}\right)^n$, and thus it goes to 0.
</s>