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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 \( s_k \geq \max \{ s_{k-1}, t - \frac{1}{k} \} \) for all \( k \).
Transcribed Image Text:**11.11** Let \( t = \sup S \). There are several ways to prove the result. (1) Provide an inductive definition where \( s_k \geq \max \{ s_{k-1}, t - \frac{1}{k} \} \) for all \( k \).
**11.11** Let \( S \) be a bounded set. Prove there is an increasing sequence \( (s_n) \) of points in \( S \) such that \( \lim s_n = \sup S \). Compare Exercise 10.7. *Note:* If \( \sup S \) is in \( S \), it's sufficient to define \( s_n = \sup S \) for all \( n \).
Transcribed Image Text:**11.11** Let \( S \) be a bounded set. Prove there is an increasing sequence \( (s_n) \) of points in \( S \) such that \( \lim s_n = \sup S \). Compare Exercise 10.7. *Note:* If \( \sup S \) is in \( S \), it's sufficient to define \( s_n = \sup S \) for all \( n \).
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