Introductory Real Analysis, Lecture 7: Monotone Convergence, Bolzano-Weierstrass, Cauchy Sequences

Real Analysis Course Lecture 7: Monotone Convergence Theorem, Bolzano-Weierstrass Theorem, Cauchy Sequence Definition and the Cauchy Convergence Criterion. Real Analysis course textbook ("Real Analysis, a First Course"): https://amzn.to/3421w9I. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP. "Hands On Start to Mathematica": https://amzn.to/2MycspH Real Analysis Playlist:    • Introduction to Real Analysis Course, Lect...   Check out my blog at: https://infinityisreallybig.com/ Follow me on Twitter:   / billkinneymath   (0:00) Exam 1 is in one week. (1:01) There's a lot to discuss in this lecture to get ready for the test. (1:44) Know definition of convergence of a sequence and be able to negate the definition and use that negated definition. (2:05) Convergent sequences are bounded: know and be able to prove. (3:06) Definition of "diverge to infinity" (be able to modify it to define what it would mean for a sequence to "diverge to minus infinity". (8:31) Algebraic properties: know, be able to prove (with help on the trickier ones), and be able to use. (14:06) The Squeeze Theorem: know, be able to prove, and be able to use. (19:12) Convergent sequences whose terms are all in a closed interval will converge to a number in that closed interval. (22:17) A monotone sequence converges iff it is bounded: know, be able to prove, be able to use. (31:04) Subsequences and the Bolzano-Weierstrass Theorem (with an aside about the proof of the monotone convergence theorem): know and be able to use (proof is harder...would require help). (36:00) Cauchy sequences and the Cauchy Convergence Criterion. Part of the proof is easy, part is hard. (45:32) A look at some homework problems that could show up on the exam. Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.