1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Compactness with accumulation points

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data

    Let K be a subset of R. Prove that if every sequence in K has an accumulation point, then K must be compact.

    2. Relevant equations

    I tried to proof it below. Am I on the right track?

    3. The attempt at a solution

    My intuition;

    Let x_n be sequence in K whose accumulation point is x, then there is a sub-sequence.
    x_n_k converges to x.
    Since sub-sequence x_n_k converges to x, x_n_k is a Cauchy sequence.
    We know that Cauchy sequences are bounded. Thus x_n is bounded. so K is bounded.

    Since all accumulation points are in K so K is closed.

    By Heine-Borel Thm, K is compact.
  2. jcsd
  3. Oct 12, 2011 #2


    User Avatar
    Science Advisor
    Gold Member

    Not quite. I think you're better off showing the equivalent (each implies the other):

    if K is not compact, there exists a sequence without an accumulation point.
  4. Oct 12, 2011 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Here you appear to be saying that if a sequence has a bounded subsequence, the sequence is bounded. Consider 1/2, 2, 1/3, 3, 1/4, 4, ...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Compactness accumulation points Date
Countably Compact Aug 25, 2017
Variations of Regular Curves problem Mar 5, 2017
Argue, why given Operators are compact or not. Nov 29, 2016
Can a compact function spaces no contain an accumulation point? Feb 5, 2013