#Math101 Session 37

Time：Apr. 2, 2026, 14:00-16:00

Location：E104 Hai Tian Building, Chang'an Campus

Speaker：Mr. Yang, Yuanxi (Class of 2024)

Title：Cauchy sequences and convergence criteria

Abstract：In this session, we continue our study of sequence convergence. We begin by introducing the monotone convergence theorem, followed by the Bolzano-Weierstrass theorem, which guarantees that every bounded sequence on the real line has a convergent subsequence. We then formalize Cauchy sequences and present the Cauchy convergence criterion, which states that a sequence of real numbers converges if and only if it is a Cauchy sequence — a property that does not necessarily hold in general metric spaces. Finally, we examine the convergence of series by interpreting them as limits of their partial sums.

（文：申爽/审核：郭千桥）