Understanding the proof of Sharkovskiis Theorem in Chaos: An introduction to dynamical systems

Question

In the book, the proof of Sharkovskiis Theorem is outlined in 11 steps but I’m stuck on step 4. Trying to give all the relevant information here to try explain my problem to someone without the book would be ridiculous, so just if anyone has the book, could you explain why step 4 part(1) must hold? Thanks

Answer 1

I would suggest the paper "A collection of simple proofs of Sharkovsky’s theorem" by Bau-Sen Du, (https://www.math.sinica.edu.tw/www/file_upload/dubs/dubs7v3.pdf). Even if you don't like the several different proofs of the theorem, then you can still go throught the huge collection of references with other proofs.