1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

General Topology

  1. Aug 29, 2007 #1
    I have the following [itex]A\subset\mathbb{R}^{n}[/itex] is dense then [itex]A[/itex] isn't bounded. Is this true? I know that [itex]A[/itex] is dense iff [itex]\bar{A}=\mathbb{R}^{n}[/itex] and that [itex]A[/itex] is bounded iff [itex]\exists \epsilon>0\mid B_{\epsilon}(0)\supset A[/itex]. How to proof it? Or there is an counterexample?
     
  2. jcsd
  3. Aug 29, 2007 #2

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Try reductio ad absurdum (proof by contradiction).
     
  4. Aug 30, 2007 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    It's slightly nicer to show that a bounded set in R cannot be dense. Many proofs by contradiction are unnecessary - i.e. you wish to show A implies B, so you assume A and not B and show not B implies not A, without any use of the assumption of A.
     
  5. Aug 30, 2007 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    But you just said "Assume A"!! Anyway, many people would consider proving the contrapositive to be "proof by contradiction".
     
  6. Aug 30, 2007 #5

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    What? The point was just make a constructive proof of the contrapositive statement without making an unnecessary preliminary assumption.
     
  7. Aug 30, 2007 #6

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Yes, I see. As I read the original post, I said to myself "Well, they both can't be true (being dense and bounded)," and this placed proof by contradiction in my mind. Bot all that is needed is ~B (bounded), so contapositive gives a direct proof.
     
  8. Aug 30, 2007 #7
    Thank you for the help!!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?