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