From an AnalExam:
Remove all numbers from the harmonic series whose denominators contain the digit 0 in their decimal representations. The resulting series converges to a number less than 90.
I believe AndrewSchoonmaker found a cute proof of this.
Unfortunately, said cute proof (a) requires assuming that the series converges and (b) often confuses people. Oh well.
Every polyhedron of genus zero has at least one face with strictly fewer than six edges.
Hint: Think GraphTheory?.