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.

For $x>0$ we have  $e^t>1$ for $0<t<x$
Hence, $$x=\int_0^x1dt \color{red}{\le}  \int_0^xe^tdt =e^x-1 \implies  1+x\le e^x$$
For $x<0$ we have  $e^{t} <1$ for $x <t<0$
$$-x=\int^0_x1dt \color{red}{\ge} \int^0_xe^tdt =1-e^x \implies  1+x\le e^x$$
