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Introductory Real Analysis, Lecture 6: Bounded Sequences, Monotone Sequences, Limits of Sequences
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Introductory Real Analysis, Lecture 6
(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.
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