Up for an amusing analysis problem?

Suppose f is a smooth real valued function defined on the real line. Prove that if for each point x there is a number N such that the nth derivative of f at x is 0 for every n >= N, then there is a nonempty interval (a, b) on which f restricts to a polynomial.

I know the solution, and it is amusing enough that I thought I would share it with the world. 10 points to the first correct answer. :)