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Nested Interval Proof:

  1. Feb 29, 2012 #1
    1. The problem statement, all variables and given/known data
    Start with the Bolzano-Weiertrass theorem and use it to construct a proof of the Nested Interval Property.
    2. Relevant equations
    Bolzano-Weiertrass: Every bounded sequence contains a convergent sub-sequence
    Nested Interval Property: Closed intervals nested inside of each other forever is non-empty.
    3. The attempt at a solution
    If we start with a bounded sequence on a closed interval and then we make it smaller we have a smaller portion of the sequence and so this smaller part must converge to something and we just keep making the interval smaller and we squeeze it down to a point, the sequence must converge to this point because it is the only point in the sequence.
    Can I just start with some interval and slowly make it approach the middle by having it increase from the right and decrease to the left till I just have enclosed one point and make it converge to this point.
     
  2. jcsd
  3. Feb 29, 2012 #2

    Deveno

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    to use B-W to prove the Nested Interval Property, you don't want to "start with a sequence". you want to start with an arbitrary family of nested closed intervals.

    can you think of a way to create a bounded sequence from such a family?

    if so, then you can say: by B-W, we know that....
     
  4. Feb 29, 2012 #3
    When we start with a family of nested closed intervals, By the B-w, we know that there should be a convergent point among these family of intervals.
     
  5. Feb 29, 2012 #4

    Deveno

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    you can only apply B-W if you have a bounded sequence. what is your bounded sequence?
     
  6. Mar 1, 2012 #5
    Do I just say I have some generic sequence A and that it is bounded between some interval.
     
  7. Mar 1, 2012 #6

    Fredrik

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    Since you're proving the nested intervals thing, you need to start with an arbitrary decreasing sequence of closed intervals. If you're going for a proof that involves sequences of real numbers, you will have to use the sequence of intervals to define a sequence of real numbers.
     
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