Sharkovsky's theorem is presented in two parts. The 'little' theorem states that a continuous function on a real interval with a periodic point of least period 3 must have periodic points of every least period. The proof uses only the intermediate value theorem and two lemmas about compact intervals. The 'great' theorem generalizes this via the Sharkovsky ordering — a total ordering on positive integers — showing that if a continuous function has a point of least period n, it must have points of every least period m where m comes after n in the Sharkovsky ordering. The ordering places odd numbers first, then their doubles, quadruples, etc., with powers of 2 in reverse order at the end.
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