Okay, last one from me tonight.
Separating hyperplane theorem and/or the Riesz extension theorem. The finite (or 2) dimensional version is fairly easy to illustrate and not too hard to prove. And of course as an example application you can assume the infinite dimensional version and derive Hahn-Banach Theorem (the version about extending linear functionals). Consider its use in convex and functional analysis, at least some of the students will run into something like this in the future.