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I Cantor's intersection theorem (Apostol)

  1. Mar 25, 2016 #1
    Hi,

    I am reading "mathematical analysis" by Apostol right now for a course in analysis. Since I am trying to understand the author's proof of the above theorem(3.25 in the book), but I have something that I can't understand.
    He assumes that each of the nested sets contains infinitely many points, "...otherwise, the proof is trivial". I can't see why it's trivial and how to prove it. I would be grateful to understand why an infinite intersection of finite sets is non-empty.

    Thank you.
     
  2. jcsd
  3. Mar 25, 2016 #2

    Samy_A

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    You have the non-empty nested sets ##C_0 ⊇ C_1 ⊇ ... ⊇ C_k ⊇ ...##.
    Assume ##C_k## is finite for some k. Then all the ##C_j## with ##j>k## will also be finite, and equal to or smaller than ##C_k##. As the number of elements in the sets of the sequence is decreasing, at a certain point in the sequence the number of elements will decrease no more (as it can't become 0).
    From that point, all ##C_j## will be equal.
    Hence the intersection will trivially not be empty
     
    Last edited: Mar 25, 2016
  4. Mar 25, 2016 #3
    Wow, thank you. What I missed was your note in the parenthesis. I feel embarrassed not noticing it.

    Thank you.
     
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