Supremum and Infimum (Sup and Inf) Definitions and Examples, Prove sup{cos(n) | n in N} Equals One

Real Analysis Study Help for Baby Rudin, Part 1.3 Key Fact: Prove sup{cos n | n ∈ ℕ} = 1. Principles of Real Analysis, by Walter Rudin, ("Baby Rudin"): https://amzn.to/39XOqzn

This is related to the facts that sup sin n = 1, limsup sin n equals 1, and limsup cos n equals 1. I define the supremum (least upper bond) and infimum (greatest lower bound) in an ordered set (usually the ordered field of real numbers), work through basic examples with closed and open intervals, as well as the range of the sequence 1 - 1/n and its union with the singleton set {1}. Finally, letting E be the range (image) of the cosine function as the inputs range over the natural numbers n = 1, 2, 3, ..., I show that the supremum of this set is one (sup cos(n) = 1). The concepts of a logical negation and a logical contrapositive are also emphasized.

#realanalysis #supremum #suprema

Infinity is Really Big Blog Post, "Study Help for Baby Rudin, Part 1.3": https://infinityisreallybig.com/2021/...

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(0:00) Introduction
(0:35) Baby Rudin and Infinity is Really Big
(0:55) The highlights of the video
(1:16) Does E have a supremum?
(2:08) Definition of the supremum
(2:56) Logical negation
(4:02) Logical contrapositive
(4:59) Example 1: E=[a,b]
(7:11) Example 2: E=(a,b)
(10:05) Examples 3a & 3b: E is a discrete infinite set
(11:52) Main Example: E={cos(n)|n in a natural number}
(12:44) Numerical evidence that sup(E)=1
(15:11) Graphical display
(16:01) Find n so that cos(n) is larger than 1-epsilon
(17:54) Understanding the proof
(19:02) Choosing delta
(19:45) Choosing N by the Archimedean property
(20:21) Computing theta_n
(21:15) Partition [0,2pi]
(22:36) Use the Pigeonhole principle
(23:49) Choosing n
(25:03) Visual of the unit circle
(25:51) Be sure to subscribe!

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