Prove using an inequality that $e$ is irrational

Question

I have to prove that $e$ is irrational using this inequality

$$0<e-\sum_{k=0}^n\frac1{k!}<\frac1{n\cdot n!}$$

The exercise leave the hint "prove by contradiction". I know too that $2<e\le 3$.

What I tried is set $e=p/q$ for $p,q\in\Bbb N$, and $\sum 1/k!=A/n!$ where $A\in\Bbb N$. Then I written

$$0<\frac{p}{q}-\frac{A}{n!}<\frac1{nn!}$$

but I dont get any idea from here. Indeed I dont know exactly what to do here, I never used a inequality to prove the irrationality of a number. Can you give me some hint (or solution)? Thank you.

P.S.: I dont know exactly what kind of tags I have to use for this question.

Answer 1

Hint: If $e=\frac{p}{q}$, take $n=q$ in the given inequality and clear denominators.