Graduate Math Meet

Title: The Arzelà–Ascoli theorem

Abstract: The Bolzano–Weierstrass Theorem guarantees that every bounded sequence of real numbers has a convergent subsequence. However, for sequences of functions, boundedness alone is not sufficient to ensure uniform convergence. In this talk, we present the Arzelà–Ascoli theorem, which gives necessary and sufficient conditions—uniform boundedness and equicontinuity—for the existence of a uniformly convergent subsequence. We also briefly explain the idea of the proof and discuss illustrative examples.