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Nested intervals, uncountable sets, unique points.

  1. Apr 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Let [a,b] be an interval and let A be a subset of [a,b] and suppose that A is an infinite set.

    Suppose that A is uncountable. Prove that there exists a point z which is an element of [a,b] such that A intersect I is uncountable for every open interval I that contains z.

    2. Relevant equations

    I don't really know how to start this problem. I know I can use the fact that a set that contains an uncountable subset is uncountable. Any help would be appreciated

    3. The attempt at a solution

    I know z is an element of I, and that I is uncountable. I also know z is an element of [an, bn]. I know there exists a unique point z in [an, bn] and I know A is uncountable.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 26, 2009 #2

    matt grime

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    Do you know about compactness?
     
  4. Apr 27, 2009 #3
    no, we have not covered compactness.
     
  5. Apr 27, 2009 #4
    mathkiddi, what "completeness" axioms/theorems have you covered? Specifically, have you covered any of these:

    least upper bound, bounded monotone sequences, Bolzano-Weierstrass, Heine-Borel, nested intervals, or anything like these? limit points? accumulation points? subsequences?

    What is [an, bn]?
     
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