The number e is irrational

Let
$$e=\sum _{k=0}^{\infty }\frac{1}{k!}=1+1+\frac{1}{2}+\frac{1}{6}+\cdots .$$

Show that $e$ is irrational.

The standard proof assumes $e=p/q$ and multiplies by $q!$. Is there a proof that does not argue by contradiction?