Prove that $e$ is irrational

I'm gonna give you one proof that is very similar to what you are looking for.

Let's give a general bound on the remainder :

$e^{x} - \sum\limits_{k=0}^{n} \frac{x^{k}}{k!} = \sum\limits_{k \geq n+1} \frac{x^{k}}{k!}$, so, getting the absolute value $\lvert \sum\limits_{k \geq n+1} \frac{x^{k}}{k!}\rvert \leq \sum\limits_{k \geq n+1} \frac{\lvert x\rvert^{k}}{k!}$

Multiply and divide by $\frac{\lvert x \rvert^{n+1}}{(n+1)!}$

$$\lvert \sum\limits_{k \geq n+1} \frac{x^{k}}{k!} \rvert = \frac{\lvert x \rvert^{n+1}}{(n+1)!} \sum\limits_{k \geq n+1} \frac{(n+1)!}{k!}\lvert x \rvert^{k-n-1} \underset{j=k-n-1}{=} \frac{\lvert x \rvert^{n+1}}{(n+1)!}\sum\limits_{j \geq 0} \frac{(n+1)!}{(n+1+j)!}\lvert x \rvert^{j}$$

Observation :

$(n+1+j)! \geq (n+1)!j! \iff \frac{(n+1+j)!}{(n+1)!j!} = \binom{n+1+j}{j} \geq 1$

So $\frac{j!(n+1)!}{(n+1+j)!} \leq 1$, because $\frac{(n+1)!j}{(n+1+j)!} = \frac{1}{\binom{n+1+j}{j}} \leq 1$

Now :

$$\lvert e^{x} - \sum\limits_{k=0}^{n} \frac{x^{k}}{k!} \rvert \leq \frac{\lvert x \rvert^{n+1}}{(n+1)!}\sum\limits_{j \geq 0} \frac{\lvert x \rvert^{j}}{j!} = \frac{\lvert x \rvert^{n+1}}{(n+1)!} e^{\lvert x \rvert}$$

Now we are able to say that $e$ is irrational.

Let's notice that rationals have the following property,for sufficiently big $n \in \mathbb{N}$,when multiplied by $n!$ they become integers,

Let's prove that $e$ doesn't respect this property :

$$n!\cdot e = [n! \sum\limits_{k=0}^{n}\frac{1}{k!}] + n!\sum\limits_{k\geq n+1} \frac{1}{k!}$$

The first term is an integer because if $k \leq n \rightarrow k! \mid n!$

For the second term let's notice given the bound just prooved that $$n! \cdot \sum\limits_{k \geq n+1} \frac{1}{k!} < n! \cdot \frac{e}{(n+1)!} = \frac{e}{(n+1)}$$

Which is not integers for $n \geq 2$ (Note that definitely belongs to $(0,1)$).

$n=1$ we can easily see that is $1 \cdot e$ would be integer it would follow that multiplied by any number would be integer, but multiplied by any $n \geq 2$ it fails due to what we've just saw.

Tell me if there's anything wrong in the proof or if there's something that doesn't feel right to you.