We’ve had at least one thread on people’s favorite equations, so it seems fair to do the same for inequalities.
Mine is the very simple 1 - x < e[sup]-x[/sup], which is used all over the place in probability theory. It’s very simple to prove, as well: 1 - x is the tangent line to e[sup]-x[/sup] at x = 0. Since e[sup]-x[/sup] is a convex function, it lies above its tangent line at any point. The result follows immediately.