1. Limited time only! Sign up for a free 30min personal 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!

Compact implies Sequentially Compact

  1. Feb 1, 2009 #1
    The problem statement, all variables and given/known data[/b]
    I need help proving that if X is a metric space and E a subset of X is compact, then E is sequentially compact.

    I know I need to consider a sequence x_n in E, and I want to say that there is a point a in E and a radius r > 0 so that Br(a) [the ball of radius r with center a] contains x_k for infinitely many k's. If I show this, then I think I can conclude that any subsequence of x_n converges to a. Can someone please help?
  2. jcsd
  3. Feb 1, 2009 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    So... your conjecture is that every sequence in a compact metric space is actally convergent?
  4. Feb 1, 2009 #3
    No, that is not one of the assumptions...my second paragraph in the problem description is a hint in the back of the textbook as to how to go about the problem.
  5. Feb 1, 2009 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If any subsequence of xn converges to a, then xn converges to a too. Rather, what you can conclude is a single subsequence of xn converges to a (which is what sequential compactness is about)
  6. Feb 16, 2009 #5
    This is not true- consider a_n=(-1)^n=1,-1,1,-1,1,-1,...

    Then a_{2n}=1 is a subsequence that converges to 1, but a_n does not converge at all- nonetheless to 1.
  7. Feb 17, 2009 #6


    User Avatar
    Science Advisor

    Office Shredder meant "if every subsequence". He chose the wrong word.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Compact implies Sequentially Compact
  1. Sequential Compactness (Replies: 8)

  2. Sequential compactness (Replies: 4)

  3. Sequential Compactness (Replies: 1)