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 simple proof...
Define function $f(x)=e^x-(x+1)$. The minimum value is $0$ at $x=0$, it's also convex, so $f(x) \ge 0$.
