Monotone subsequence via ultrapower

TL;DR

This paper explores the use of ultraproducts to analyze the existence of monotone subsequences in infinite sequences, providing a novel perspective on classical calculus problems and related applications.

Contribution

It introduces a new approach using ultrapowers to prove the monotone subsequence theorem and demonstrates its applicability to other problems in ordered structures.

Findings

Ultraproducts can be effectively used to prove classical theorems in analysis.

The method provides a unifying framework for understanding saturation and compactness in ordered structures.

Applications extend beyond the monotone subsequence problem to broader areas in mathematics.

Abstract

An ultraproduct can be a helpful organizing principle in presenting solutions of problems at many levels, as argued by Terence Tao. We apply it here to the solution of a calculus problem: every infinite sequence has a monotone infinite subsequence, and give other applications. Keywords: ordered structures; monotone subsequence; ultrapower; saturation; compactness