In class today we proved that $\Bbb{Q}$ is not complet, you used the fact that $$ \sum_{k=0}^N\frac{1}{k!}\underset{N\to+\infty}{\longrightarrow}e\notin\Bbb{Q}.$$
After that I was perplex to prove that $e\notin\Bbb{Q}$, the method I know used the fact that $e=\displaystyle{\sum_{k=0}^{+\infty}}\frac{1}{k!}$. It seems weird (here) to use this method. So my question is :
Can I use this method to prove that $e$ is irrational? At least I would like to have a mathematical argument.
Remark: Sure, we can always change the sequence and choose for example $u_n=\bigl(1+\frac{1}{n}\bigr)^n$.
It may be much easier for you to prove that $\sqrt{2}$ is irrational. (I know it is for me.) Anyway there are many rational series that converge to $\sqrt{2}$. My favorite is: $$\sqrt{2} = 1+ \frac{1}{2+\frac{1}{2+\frac{1}{2+\ddots}}} $$