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

Arzelà's Bounded Convergence Theorem (1885) states that if a sequence of Riemann integrable functions on a compact interval is uniformly bounded and has an integrable pointwise limit function, then the sequence of their integrals tends to the integral of the limit. It is a trivial consequence of measure theory; in particular, of the Dominated Convergence Theorem. However, denying oneself the machinery of measure theory transforms this simple and intuitive result into a surprisingly di cult problem. Indeed, despite the naturality, applicability, and elegance of the result, the proof is omitted from most introductory analysis texts. Here, we present a novel argument suitable for consumption by freshmen.