Introductory Real Analysis, Lecture 6: Bounded Sequences, Monotone Sequences, Limits of Sequences
Learn about Bounded Sequences, Monotone Sequences, and Limits of Sequences. 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 Introductory Real Analysis, Lecture 6 Real Analysis Playlist: • Introduction to Real Analysis Course, Lect... Check out my blog at: https://infinityisreallybig.com/ Follow me on Twitter: / billkinneymath (0:00) Recursively defined sequence exhibiting chaotic behavior. (3:40) Describe the butterfly effect. (8:29) Definition of a bounded sequence. (12:56) Definition of a monotone increasing sequence. (16:21) You should try to come up with a definition of "eventually monotone increasing" on your own. (17:22) Definition of a limit of a sequence. (21:48) Limit notation. (22:55) Scratch work and proof that {3n/(5n+11)} converges to 3/5. (35:51) Negate the definition of convergence of a sequence and do scratch work for confirming that {3n/(5n+11)} does not converge to 0.60001. (46:15) Summarize other things: convergent sequences are bounded (with discussion of converse and contrapositive) and the limit of a convergent sequence is unique, discussion of what it means to "diverge to infinity". (50:17) Prove that the limit of the sum of two convergent sequences exists and is the sum of the individual limits. Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.
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