The ultimate periodicity theorem is an important result in min-max systems theory. It was first proved by Olsder and Perennes in their unpublished work. In this note, we present a new proof. This proof is also based on two important theorems: the existence of cycle time for any min-max function and the Nussbaum-Sine theorem. However, two different techniques, pure min-max function and conditional redundancy, are used to obtain two important intermediate results. The purpose of this note is to provide a simple alternate proof to the ultimate periodicity theorem.