2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/137482Based on various strategies, we obtain several simple proofs of the celebrated Sharkovsky cycle coexistence theorem.20 pages. New proofs involving Stefan cycles are included in sections 8 and 9. New proofs of (a) and (c) are included in sections 11 and 12Dynamical Systems37E05text