real analysis - Trying to show that $e-sum_{k=0}^{n}1/k!$ goes to $0$ faster than $n!$ goes to infinity as $nto infty$

This have been asked before but I think people misunderstood my question.

For a better notation:

$e-\sum_{k=0}^{n}1/k! = e - \sum^n$ .

Having the following inequality:

$0 < n!(e-\sum^n) <1/n \tag{1}$

we can apply the squeeze theorem to show that $n!(e-\sum^n)$ goes to zero as $n$ goes to infinity.

If $n!(e-\sum^n)$ goes to zero when $n \to \infty$ this means that $\sum^n$ converges so quickly to $e$ that $(e-\sum^n)$ goes to $0$ faster than $n!$ goes to infinity as $n \to \infty$.

Is possible to show this without relying on $(1)\:?$