Compactness Arguments in Real Analysis

TL;DR

This paper demonstrates that elementary real analysis theorems can be proved using four simple types of compactness arguments, avoiding more complex notions and emphasizing the sufficiency of basic concepts.

Contribution

It introduces four independent approaches to compactness arguments in real analysis, simplifying proofs and clarifying foundational methods.

Findings

Four types of compactness arguments are sufficient for elementary real analysis proofs.

Simplified proofs avoid advanced notions beyond basic real number properties.

The approach enhances understanding of core concepts in real function theory.

Abstract

Theorems crucial in elementary real function theory have proofs in which compactness arguments are used. Despite the introduction in relatively recent literature of each new highly elegant compactness argument, or of an equivalent, this work is based on the idea that, with the aid of simple notions such as local properties of continuous or of differentiable functions, suprema, nested intervals, convergent subsequences or the simplest form of the Heine-Borel Theorem, the use of one of four simple types of compactness arguments, suffices, and the resulting development of real function theory need not involve notions more sophisticated than what immediately follows from the usual ordering of the real numbers. Thus, four independent approaches are presented, one for each type of compactness argument: supremum arguments, nested interval arguments, Heine-Borel arguments and sequential compactness arguments.