Prove that the closure of a bounded set is bounded.

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SUMMARY

The discussion centers on proving that the closure of a bounded subset S of ℝ^n is also bounded. Participants emphasize the importance of understanding the definitions of boundedness and closure, as well as the concept of open balls. One contributor points out a common mistake in proof formulation regarding the quantifier used with epsilon, suggesting it should be "there exists epsilon" instead of "for all epsilon." Additionally, the discussion touches on the relationship between closure and closed subsets, although this result has not yet been formally proven in the context of the discussion.

PREREQUISITES
  • Understanding of bounded sets in ℝ^n
  • Familiarity with the concept of closure in topology
  • Knowledge of open balls and their properties
  • Basic proficiency in mathematical proof techniques
NEXT STEPS
  • Study the definitions and properties of closed sets in topology
  • Learn about the concept of closure as the intersection of closed subsets
  • Review the formal proof techniques in real analysis
  • Examine examples of bounded and unbounded sets in ℝ^n
USEFUL FOR

Mathematics students, particularly those studying real analysis and topology, as well as educators looking to clarify concepts related to boundedness and closure in mathematical proofs.

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.
 

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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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