Simplest or nicest proof that $1+x \le e^x$ The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which fits into a few lines, is accessible to students with limited calculus experience, and does not involve too much analysis of different cases.
Another way (not sure if its "simple" though!):  $y = x+1$ is the tangent line to $y = e^x$ when $x= 0$.  Since $e^x$ is convex, it always remains above its tangent lines.
$1+x \le e^x$ 

Take the ln of both sides

$\ln(1+x) \le x$ 

differentiate both sides w.r.t $x$

$\frac{1}{1+x} \le 1$

which holds $\forall x \in \mathbb{R}, x \neq -1$.