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5. a.) Consider the collection of open sets {(-n, n) ³. in R with the usual metric d(x,y) = /x-yl. Use this collection show IR is not compact. (make R=Unes (-n, n) as part of your work) b. Give an example of a collection of compact 1R, say {Kn³0. s.t. part (a.)) open sets to to prove Sure sets in Uns Kn is not compact. (hint: consider of

5. a.) Consider the collection of open sets {(-n, n) ³. in R with the usual metric d(x,y) = /x-yl. Use this collection show IR is not compact. (make R=Unes (-n, n) as part of your work) b. Give an example of a collection of compact 1R, say {Kn³0. s.t. part (a.)) open sets to to prove Sure sets in Uns Kn is not compact. (hint: consider of

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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**Problem 5:** a) Consider the collection of open sets \(\{(-n, n)\}_{n=1}^{\infty}\) in \(\mathbb{R}\) with the usual metric \(d(x, y) = |x - y|\). Use this collection of open sets to show \(\mathbb{R}\) is not compact. (Make sure to prove \(\mathbb{R} = \bigcup_{n=1}^{\infty} (-n, n)\) as part of your work.) b) Give an example of a collection of compact sets in \(\mathbb{R}\), say \(\{K_n\}_{n=1}^{\infty}\), such that \(\bigcup_{n=1}^{\infty} K_n\) is not compact. (Hint: consider part (a).)
Transcribed Image Text:**Problem 5:** a) Consider the collection of open sets \(\{(-n, n)\}_{n=1}^{\infty}\) in \(\mathbb{R}\) with the usual metric \(d(x, y) = |x - y|\). Use this collection of open sets to show \(\mathbb{R}\) is not compact. (Make sure to prove \(\mathbb{R} = \bigcup_{n=1}^{\infty} (-n, n)\) as part of your work.) b) Give an example of a collection of compact sets in \(\mathbb{R}\), say \(\{K_n\}_{n=1}^{\infty}\), such that \(\bigcup_{n=1}^{\infty} K_n\) is not compact. (Hint: consider part (a).)
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