The Sharkovsky Theorem

TL;DR

This paper presents the original, detailed proof of the Sharkovsky theorem, emphasizing its pedagogical value and the auxiliary results that may interest researchers, despite the existence of alternative proofs.

Contribution

It provides the first comprehensive, accessible presentation of the original proof, including auxiliary results that are of independent interest.

Findings

Original proof detailed and accessible

Auxiliary results of independent interest

Clarifies the path of the original derivation

Abstract

The original proof of the Sharkovsky theorem is presented in full detail. The proof should be accessible to readers with basic Real Analysis background. Although nowadays there are several alternative proofs of this classical result, we believe the original proof has independent interest. On the one hand, it shows the path followed by the author to derive his result; on the other hand, many auxiliary results in this proof have independent interest. Moreover, the proof itself is not widely available, is very terse, and contains several typographical errors.