# Sharkovsky's theorem, period 4 implies period 2

I need to prove that using the ordering of the Sharkocsky's, that period 4 implies period 2. Thus for a continuous function f from the unit interval to the unit interval itself, I need to prove that $$f^4(x)=x$$ implies $$f^2(y)=y$$. But I do not know how to do this. Can somebody help me with this proof. I also had to prove that period 2 implies period 1, but that can be done with the intermediate value theorm.

Let $$g=f^2$$. Then $$g$$ has a periodic point of period $$2$$. Can you finish?
• You need also to exclude that the 1-periodic point of $f^2$ that you get is not the 1-periodic point of $f$. – LutzL Oct 8 '18 at 17:53