Compactness Arguments in Real Analysis

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.