$e$: $\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n$ = $\lim\limits_{n \to \infty}\sum\limits_{i=0}^n\left(\frac{1}{i!}\right)$

$\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n$ = $\lim\limits_{n \to \infty}\sum\limits_{i=0}^n\left(\frac{1}{i!}\right)$ since:

(1) Since, it can be shown that both  $\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n$ and $\lim\limits_{n \to \infty}\left(\frac{1}{n!}\right)$, the only remaining point is to show that:

$$\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n =\lim\limits_{n \to \infty}1 + 1 + \frac{1}{2!}\left(1 - \frac{1}{n}\right) + \frac{1}{3!}\left(1 - \frac{1}{n}\right)\left(1 - \frac{2}{n}\right) + \dots +$$ $$+ \frac{1}{n!}\left(1 - \frac{1}{n}\right)\left(1 - \frac{2}{n}\right)\dots\left(1 - \frac{n-1}{n}\right)  = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots + \frac{1}{n!}$$