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Nested interval theorem for R^n

  1. May 2, 2005 #1
    Hi,

    I am trying to prove the nested interval theorem for R^n. It is stated as follows:

    Let (K_i : i in N) be a decreasing sequence of bounded closed sets in R^n and each K_i is non empty. Then the intersection of all K_i for i in N is not empty.

    This is what I have so far:

    Since K_i is nonempty, there exists a p_i in K_i for i in N. Pick these points to form the sequence (p_i : i in N). Because (p_i : i in N) is bounded, it has at least one convergent subsequence by the Bolzano-Weierstrass theorem.

    I want to claim that the limit point of this subsequence is in the intersection of all K_i. But I am having trouble formalizing this part of the argument. Any help?
     
  2. jcsd
  3. May 2, 2005 #2
    The key is these sets are nested. So [tex]p_i \in K_i \subset K_{i-1} \subset K_{i-2} \subset \ldots \subset K_1[/tex]

    So for [tex]j\ge i[/tex], [tex]p_j \in K_i[/tex]. So the tail of the convergent subsequence is in [tex]K_i[/tex]. So the limit in is [tex]K_i[/tex] since it is closed. Since this is true for an arbitrary i it is true for all i. So the limit is in all of the [tex]K_i[/tex]'s so it is in the intersection.

    Hope that helps,
    Steven
     
  4. May 2, 2005 #3
    Excellent, many thanks.
     
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