A Concise, Elementary Proof of Arzela's Bounded Convergence Theorem

Arzela'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 point-wise 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 Arzela 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.