Prove $ 2<(1+\frac{1}{n})^{n}$

Hint: Use the binomial theorem, look at the first two to three terms.

you can use Bernoulli's inequality

for every $ x>-1$ $$\\ { \left( 1+x \right) }^{ n }\ge 1+nx$$

To show $\Big( 1+\frac{1}{n}\Big)^n>2\Longleftrightarrow$ To show $(n+1)^n>2n^n$

Note that, for $n>1$, $$(n+1)^n=\sum_{k=0}^{n}\binom{n}{k}n^k>n^n+\binom{n}{1}n^{n-1}=n^n+n^n=2n^n\space\space\space\blacksquare$$