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This note is intended primarily for college calculus students right after theintroduction of the Intermediate Value Theorem, to show them how theIntermediate Value Theorem is used repeatedly and straightforwardly to provethe celebrated Sharkovsky's theorem on the periods of coexistent periodicorbits of continuous maps on an interval. Furthermore, if the maps have aperiodic orbit P of odd period > 1, then we find more periodic points, in theappendix, which constitute what we call the towers of periodic pointsassociated with P. These towers of periodic points have infinitely many layers.In this note, no knowledge of Dynamical Systems Theory is required.  In this revision, we add the result (Proposition 1 on page 9) that theperiodic orbits of continuous unimodal maps on the interval $[0, 1]$ of leastperiods $\ge 2$ are nested in the sense that if $P$ and $Q$ are periodic orbitsof a continuous unimodal map on $[0, 1]$ of least periods $\ge 2$ and if $\maxP < \max Q$, then $[\min P, \max P] \subset [\min Q, \max Q]$.

An Interesting Application of the Intermediate Value Theorem: A Simple Proof of Sharkovsky's Theorem and the Towers of Periodic Points