Compactness Arguments in Real Analysis
Rafael Cantuba
TL;DR
The paper addresses proving core real analysis results using four elementary compactness arguments, avoiding heavy topology by emphasizing simple notions like suprema, nested intervals, convergent subsequences, and the Heine-Borel form. It restates key theorems in primed forms (IVT', DIT', SIFT', IFT', MVI', CFT', I1') and develops four unified proof strategies: supremum arguments, nested-intervals, Heine-Borel arguments, and sequential-compactness arguments. It then demonstrates how these approaches yield fundamental results such as the Intermediate Value Theorem, Darboux integrability, the Mean Value Theorem, and the Fundamental Theorem of Calculus, illustrating a coherent, locally grounded path to elementary real function theory. The framework emphasizes modularity and intuition, offering flexible pedagogical options that connect basic real-number properties with global analytic conclusions via compactness without heavy topological machinery.
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.