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?