real analysis - limit of $left( 1-frac{1}{n}right)^{n}$

limit of

$$\left( 1-\frac{1}{n}\right)^{n}$$

is said to be $\frac{1}{e}$ but how do we actually prove it?

I'm trying to use squeeze theorem

$$\frac{1}{e}=\lim\limits_{n\to \infty}\left(1-\frac{1}{n+1}\right)^{n}>\lim\limits_{n\to \infty}\left( 1-\frac{1}{n} \right)^{n} > ??$$