Prove $ 2<(1+\frac{1}{n})^{n}$ How to prove that $ 2<(1+\frac{1}{n})^{n}$ for every integer $n>1$. I was thinking by induction, it works for $n=2$ but then I couldn't move forward.

Approach using AM-GM Inequality
First we show that 
$$f(n)=\left(1+\frac{1}{n}\right)^n$$
is monotone increasing. To see this, apply the AM-GM inequality to the following $n+1$ terms
$$\left\{1,1+\frac{1}{n},1+\frac{1}{n},\cdots n\mbox{ times}\right\}$$
You get
$$\frac{1+n+1}{n+1} \geq \left(1+\frac{1}{n}\right)^{\frac{n}{n+1}}$$
Rearranging gives you $f(n) < f(n+1)$ since the terms in AM-GM are not equal.
Since $f(1)=2$, the result follows.
