Elementary real analysis without compactness argument

TL;DR

This paper revisits classical real analysis results, presenting proofs based on the key-lemma behind Cousin's theorem instead of traditional compactness arguments.

Contribution

It introduces a new proof approach for elementary real analysis theorems using Cousin's theorem's key-lemma, avoiding compactness.

Findings

Simplifies proofs of classical theorems

Highlights the utility of Cousin's theorem in analysis

Provides alternative methods for foundational results

Abstract

In a famous paper, R. A. Gordon proved a dozen theorems using tagged partitions and Cousin's theorem. The purpose of this paper is to present several classical results using the key-lemma underlying Cousin's theorem.