Prove that $\left(1+\frac1 n\right)^n > 2$ I'm trying to demonstrate that $\left( 1+\frac1 n \right)^n$ is bigger than $2$. I have tried to prove that $\left( 1+\frac1 n \right)^n$ is smaller than $\left( 1+\frac1{n+1} \right)^{n+1}$ by expanding 
$\left( 1+\frac1n \right)^n = \sum\limits_{i=0}^n \left( \frac{n}{k} \right) \frac{1}{n^k}$ and $\left( 1+\frac1{n+1} \right)^{n+1} = \sum\limits_{i=0}^{n+1} \left( \frac{(n+1)}{k} \right) \frac{1}{(n+1)^k}$ but it doesn't seem to work.
What am I missing? Also, is there a method to demonstrate that without induction?

Another way is to prove first that your sequence is monotonically increasing like has been done here:
I have to show $(1+\frac1n)^n$ is monotonically increasing sequence
... and since your first term is $2$, it follows that the subsequent ones are larger than $2$.
