Prove that the closure of a bounded set is bounded.

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Homework Help Overview

The discussion centers around proving that the closure of a bounded subset of ℝ^n is also bounded. The original poster presents a statement and definitions related to boundedness and closure.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to provide a proof based on definitions, while some participants suggest alternative approaches and question the necessity of certain definitions that have not yet been established in their studies.

Discussion Status

Participants are actively engaging with the original proof, offering feedback on clarity and correctness. There is acknowledgment of the original proof's validity, but also a recognition of the need for further foundational understanding regarding the closure of sets.

Contextual Notes

There is a mention of typos in the original proof and a specific concern about the phrasing of a condition involving epsilon. Additionally, the original poster notes that they have not yet proved a related result about the closure of sets, which affects their ability to use it in their proof.

dustbin
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Homework Statement



Prove that if S is a bounded subset of ℝ^n, then the closure of S is bounded.

Homework Equations



Definitions of bounded, closure, open balls, etc.

The Attempt at a Solution



See attached pdf.
 

Attachments

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That looks OK to me apart from a couple of typos.

There is an easier way to prove this too though. Do you know that the closure of S is the intersection of all closed subsets containing S?
 
Thanks for your input! We have not proved that the closure of S is the intersection of all closed subsets containing S... so I cannot use this result without proof.
 
dustbin said:
Thanks for your input! We have not proved that the closure of S is the intersection of all closed subsets containing S... so I cannot use this result without proof.

Your original proof looks fine to me. Except when you pick an epsilon, it should be "there exists epsilon" not "for all epsilon".
 

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