Suppose that a continuous map of the interval to itself has a periodic point that is not fixed. Then it must have a periodic point whose least period is exactly two. This result is interesting in its own right and is used in the proof of Sharkovskii's theorem (of which it is a special case). The proof that we give is simple enough to be used in undergraduate courses on dynamical systems.
This article has appeared in the American Mathematical Monthly 107(2000), 932--933. It is available in the following formats:
Authors' addresses: Reid Barton 66 Alpine Street Arlington MA 02474 U.S.A. Keith Burns Department of Mathematics Northwestern University Evanston, IL 60208-2730 U.S.A. burns followed by math.northwestern.edu