analysis - Convergence of Sequence with factorial

I want to show that
$$
a_n = \frac{3^n}{n!}
$$
converges to zero. I tried Stirlings formulae, by it the fraction becomes

$$
\frac{3^n}{\sqrt{2\pi n} (n^n/e^n)}
$$
which equals
$$
\frac{1}{\sqrt{2\pi n}} \left( \frac{3e}{n} \right)^n
$$
from this can I conclude that it goes to zero because $\frac{3e}{n}$ and $\frac{1}{\sqrt{2\pi n}}$ approaching zero?