Any Goodstein's sequence
(ie. starting with any n and b positive integers) converges back to zero in a finite number of steps.
Kirby and Paris proved that this theorem can only be proved using the concept of an infinite number and thus proved the necessity of the concept of infinity in mathematics.
It's an example of a Godel's theorem for induction. Given the correctness of mathematical induction, we
must believe Goodstein's theorem even though it cannot be proved by mathematical induction.