The elementary but very useful inequality that 1+x≤ex 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.

**Answer**

Another way (not sure if its “simple” though!): y=x+1 is the tangent line to y=ex when x=0. Since ex is convex, it always remains above its tangent lines.

**Attribution***Source : Link , Question Author : Ashley Montanaro , Answer Author : Macavity*