Sharkovsky's theorem, period 4 implies period 2

Question

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.

Answer 1

Let $g=f^2$. Then $g$ has a periodic point of period $2$. Can you finish?