MIT 18.100B Real Analysis, Spring 2025 - Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series

MIT 18.100B Real Analysis, Spring 2025 Instructor: Tobias Holck Colding View the complete course: https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/ YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62Ie7F_tTAhhXoX5_Cl8meG The Bolzano-Weierstrass theorem say that any bounded sequence has a convergent subsequence. This crucial fact can be used to show other important theorems. An almost immediate consequence of it is the Cauchy convergence theorem. We also introduce the notion of a series and get acquainted with perhaps the single most important series: the geometric series. For series, the most important question is whether or not it is convergent. To determine that, there are a number of tests beginning with the two versions of the comparison test. License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu Support OCW at http://ow.ly/a1If50zVRlQ We encourage constructive comments and discussion on OCW’s YouTube and other social media channels. Personal attacks, hate speech, trolling, and inappropriate comments are not allowed and may be removed. More details at https://ocw.mit.edu/comments.