A Concise, Elementary Proof of Arzel\`a's Bounded Convergence Theorem

TL;DR

This paper provides a simple, elementary proof of Arzelà's bounded convergence theorem, making it accessible to beginners and clarifying a classical result often omitted from basic analysis courses.

Contribution

It introduces a new, straightforward proof of Arzelà's theorem that avoids measure theory, suitable for teaching undergraduates.

Findings

Proof is accessible to freshmen

Clarifies the classical theorem's validity without advanced machinery

Highlights the theorem's naturality and importance in analysis

Abstract

Arzel\`a's bounded convergence theorem (1885) states that if a sequence of Riemann integrable functions on a closed interval is uniformly bounded and has an integrable pointwise limit, then the sequence of their integrals tends to the integral of the limit. It is a trivial consequence of measure theory. However, denying oneself this machinery transforms this intuitive result into a surprisingly difficult problem; indeed, the proofs first offered by Arzel\`a and Hausdorff were long, difficult, and contained gaps. In addition, the proof is omitted from most introductory analysis texts despite the result's naturality and applicability. Here, we present a novel argument suitable for consumption by freshmen.