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A tiny topological claim in a larger proof of mine

  1. Apr 6, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm writing a proof for my Real Analysis III class, and in one clause I claim that the intersection of my countably infinite set of intervals {En} where En=(1+1/2+1/3+1/4+....+1/n , ∞), has the property that the infinite intersection of all En's equals ∅ (This would be a lot easier to explain if I finally took the time to learn Latek).

    2. Relevant equations

    Not many.

    3. The attempt at a solution

    Obviously, since ∑1/k ---> ∞, my intersection of my set of intervals gets infinitely smaller in length, ultimately approaching (∞, ∞)=∅. I'd like someone to explain this in formal topological language since I'm not taking another topology class this summer.
     
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  3. Apr 6, 2012 #2

    Fredrik

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    Two sets are equal if and only if they have the same members, so you need to prove that this intersection has no members. Let x be an arbitrary real number. Can you prove that x is not a member of this intersection?

    The LaTeX guide for the forum can get you started with LaTeX very quickly.
     
  4. Apr 6, 2012 #3
    Obviously. Since Ʃ(1/k)=1+1/2+1/3+...+1/n ---> ∞, for any x in (1+1/2+1/3+...+1/n, ∞) there is an N≥n large enough that 1+1/2+1/3+...+1/N ≥ x, and consequently (1+1/2+1/3+...+1/n, ∞) & (1+1/2+1/3+...+1/N, ∞) = (1+1/2+1/3+...+1/N, ∞) does not contain contain x.
     
  5. Apr 6, 2012 #4

    Fredrik

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    Right, there's an n such that ##x\notin\big(\sum_{k=1}^n\frac 1 k,\infty\big)##. (Not sure why you started with an x in that interval though. That makes the proof a bit weird. Just let x be an arbitrary real number). Do you also understand why this implies that ##x\notin\bigcap_{m=1}^\infty\big(\sum_{k=1}^m \frac 1 k,\infty\big)##, and why that implies that ##\bigcap_{m=1}^\infty\big(\sum_{k=1}^m \frac 1 k,\infty\big)=\emptyset##?
     
    Last edited: Apr 6, 2012
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