A very short proof of $e$ is irrational I was talking with my friend and he came up with this very short proof

Given $x\in \mathbb{R}$, if $xy \notin \mathbb{Z}$ for any $y\in \mathbb{Z}$, then $x$ is irrational. Since $e = \sum \frac{1}{n!}$, we see that $ey \notin \mathbb{Z}$ for any $y \in \mathbb{Z}$. So $e$ is irrational.

Is his argument correct? If not, why?

Perhaps you can take the fractional part of $n! \times e$  that number is
$$ 0< n! e - \text{whole number } = \frac{1}{n+1}
+ \frac{1}{(n+1)(n+2)}+\dots <1$$
if $e$ is rational  this number  has to be integer eventually. However, this error term is strictly between $0$ and $1$ always. So $e \notin \mathbb{Q}$.
https://en.m.wikipedia.org/wiki/Proof_that_e_is_irrational
