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Proof about sequence properties

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Use the Monotone convergence theorem to give a proof of the Nested interval property.
    2. Relevant equations
    Monotone convergence theorem: If a sequence is increasing or decreasing and bounded then it converges.
    Nested Interval property: If we have a closed interval [a,b] and we keep making intervals inside this and they keep getting smaller the union of all these intervals is non-empty and contains one element.
    3. The attempt at a solution
    If we started at the left endpoint of some closed interval and we had a monotonically increasing sequence and it continued on the to right with equally spaced steps, and we had a decreasing sequence that started from the right endpoint, eventually these 2 sequences will be heading towards each other and eventually reach the same common point. I think I need to be careful about how I pick the spacing between the terms in the sequence. Am I headed in the right direction with this.
     
  2. jcsd
  3. Feb 28, 2012 #2

    micromass

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    You're doing ok. You indeed have to consider the sequence of end-points. This proves that there is an element in all the nested intervals.
     
  4. Feb 29, 2012 #3
    Ok so lets say we have a closed interval [a,b] and we have a sequence that is increasing and it starts at a and steadily goes up to b in equal steps. Now lets take this same sequence and start it at b and run it backwards b going down to a. Now if we start at a and go up to b, and each time we move from a we also move from b so we have elements that are common to all intervals because they are enclosed inside each other because the sequence is steadily increasing and decreasing. Now we have to consider where the 2 endpoints are getting close to each other, these 2 sequences will eventually reach the same point because we constructed it from the same sequence. So there will be a point that is common to all intervals. lets assume that there wasn't a point that was common to all intervals. This would imply that there was a point where the 2 sequences jumped passed each other, but this couldn't happen because the 2 sequences would have to reach the same point because we constructed them this way to make that happen.
     
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