Induction on $(1+\frac{1}{n})^n \geq 2$ $$(1 + \frac{1}{n})^n \geq 2$$
Base base: $n = 1$
LHS: $(1+ 1)^1 \leq 2$  RHS: 2
$2 \leq 2$ True. 
Inductive step:
$$(1 + \frac{1}{n+1})^{n+1} \geq 2$$
$$(1 + \frac{1}{n+1})^n (1 + \frac{1}{n+1}) \geq 2$$
Im stuck 

Hint: Use Bernoulli's inequality:
$$
(1+x)^n \ge 1 + nx \quad \mbox{for $x\ge 0$}
$$
This can be proved by induction.
