Simplest or nicest proof that 1+x≤ex1+x \le e^x

The elementary but very useful inequality that 1+xex 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=ex when x=0. Since ex is convex, it always remains above its tangent lines.

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

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