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Let S be a bounded set. Prove there is an increasing sequence (sn) of points in S such that lim sn = sup S. Compare Exercise 10.7. Note: If sup S is in S, it's sufficient to define sn = sup S for all n.

Let S be a bounded set. Prove there is an increasing sequence (sn) of points in S such that lim sn = sup S. Compare Exercise 10.7. Note: If sup S is in S, it's sufficient to define sn = sup S for all n.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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my question is 11.11 in picture, another is hint .

please use （1）in the hint, you should use “contruction by induction”

 

10.7 : Let S be a bounded nonempty subset of R such that sup S is not in S. Prove there is a sequence (sn) of points in S such that lim sn = supS.

11.11 Let t = sup S. There are several ways to prove the result. (1) Provide an inductive definition where Sk≥ max{sk-1,t - } for all k. k
Transcribed Image Text:11.11 Let t = sup S. There are several ways to prove the result. (1) Provide an inductive definition where Sk≥ max{sk-1,t - } for all k. k
11.11 Let S be a bounded set. Prove there is an increasing sequence (sn) of points in S such that lim sn = sup S. Compare Exercise 10.7. Note: If sup S is in S, it's sufficient to define sn sup S for all n. =
Transcribed Image Text:11.11 Let S be a bounded set. Prove there is an increasing sequence (sn) of points in S such that lim sn = sup S. Compare Exercise 10.7. Note: If sup S is in S, it's sufficient to define sn sup S for all n. =
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