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Examples of taking limits of intervals

  1. Jul 9, 2008 #1
    I guess I've fallen through some of the cracks in the plethora of definitions I've learned, or I just never had enough examples of taking limits of intervals. Anyways, which is true, and why?

    I think the first.

    However, if I were to write the following instead:
    [tex]$\lim _{n \rightarrow \infty} \cap_{n}(0,\frac{1}{2^{n-1}}]=\varnothing$[/tex],
    would I be correct?

    If not, why?

    Last edited: Jul 9, 2008
  2. jcsd
  3. Jul 9, 2008 #2


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    Re: [tex]$\cap^{\infty}_{n=1}(0,\frac{1}{26{n-1}}]=0?$[/tex]

    No, the first doesn't even make sense- on the left you have a set and on the right you have a number. If you meant
    [tex]\cap^{\infty}_{n=1}(0,\frac{1}{2^{n-1}}]=\{ 0 \}[/tex]
    That would make sense (both sides of the equation are sets) but would also be incorrect. The notation (0, x] means "all points between 0 and x, including x but not 0". 0 is not in any of those intervals and so cannot be in their intersection. It is the second that is correct.

    Yes, the second is just definition of
    and is exactly the same as the second of your first two equations.
    Last edited by a moderator: Jul 9, 2008
  4. Jul 9, 2008 #3
    Re: [tex]$\cap^{\infty}_{n=1}(0,\frac{1}{26{n-1}}]=0?$[/tex]


    I'm trying to show that for [tex]F_{n} \subset F_{n+1}[/tex] for all n (they're sigma-algebras), that [tex]\cap ^{\infty}_{i=1}F_{i}[/tex] may not be a sigma algebra. Counter example is what the exercise says.

    I was thinking of a sequence of intervals that never include 0. But I didn't know how to properly express it such that the intersection never included 0. I didn't want to include any epsilons, but I it seems I have to. Here's what I have now.


    Will that suffice, or is there a better example?
  5. Jul 9, 2008 #4


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    Re: [tex]$\cap^{\infty}_{n=1}(0,\frac{1}{26{n-1}}]=0?$[/tex]

    I don't understand what you are trying to do. A sigma algebra is a collection of sets (that is closed under unions and complements). You are saying that you want to give an example of a collection of sigma algebras whose intersection is not a sigma algebra but your example is a single collection of sets (NOT a sigma algebra) and you are taking the intersection of the sets in that single collection of sets.
  6. Jul 9, 2008 #5
    Re: [tex]\cap^{\infty}_{n=1}(0,{1 \over 2^{n-1}}]=0?[/tex]

    As far as I know you don't have to think that intersection as a limit just because the [tex]\infty[/tex] symbol appears.
    Whenever you have a NOT EMPTY collection of sets [tex]\cal A[/tex] (for a collection of sets I mean a set [tex]\cal A[/tex] whose elements are sets themselves... for istance, given a set [tex]A[/tex], the set [tex]{\cal P}(A)[/tex] of the subsets of [tex]A[/tex] is a collection of sets) you can define the [tex]\bigcap {\cal A}[/tex] as the following set.

    [tex]\bigcap {\cal A} = \left\{ x \in Y\, |\,\, \forall X \in {\cal A}, \ x \in X \right\}[/tex]

    where [tex]Y[/tex] is any element of the collection of sets(that is non empty).

    The definition do not depend on the choice of Y and you don't require any idea of convergence or limit.
    Last edited: Jul 9, 2008
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