Math Help - Proof involving e

1. Proof involving e

How do I show that for every x>0 on has
(1+x)^(1/x)<e
I proved that e is the limit of the equation, but I don't know how to prove it is strictly less than.

e being defined as the limit as x tends to infinity of (1+1/x)^x.

2. If $f(x) = (1+x)^{1/x}$ is strictly decreasing on $x>0$ and approaches $e$ as $x \rightarrow 0$, then it must be less than $e$ for $x>0$. So I think showing that $f'(x) < 0$ for $x>0$ should suffice.