Prove that the closure of a bounded set is bounded.

  • Thread starter dustbin
  • Start date
  • #1
dustbin
240
5

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

  • Problem.pdf
    73.3 KB · Views: 1,196

Answers and Replies

  • #2
clamtrox
938
9
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?
 
  • #3
dustbin
240
5
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.
 
  • #4
Dick
Science Advisor
Homework Helper
26,263
621
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".
 

Suggested for: Prove that the closure of a bounded set is bounded.

  • Last Post
Replies
18
Views
653
Replies
4
Views
548
Replies
3
Views
362
Replies
5
Views
380
Replies
2
Views
287
Replies
7
Views
478
Replies
6
Views
879
  • Last Post
2
Replies
43
Views
2K
Replies
7
Views
1K
Replies
16
Views
339
Top