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Math Help - Proof convergence sequence of sets

  1. #1
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    Proof convergence sequence of sets

    hey

    i've got some troubles with this one:


    A sequence of sets is i monotone if A_n \subseteq A_{n+1} respectively A_n \supseteq A_{n+1} .

    Show that every monotonic sequence of sets converged and calulate the limes.

    I know that if it converges the lim sup equals the lim inf but how do i show that?

    thx
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  2. #2
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    Quote Originally Posted by hiddy View Post
    A sequence of sets is i monotone if A_n \subseteq A_{n+1} respectively A_n \supseteq A_{n+1} .

    Show that every monotonic sequence of sets converged and calulate the limes.

    I know that if it converges the lim sup equals the lim inf but how do i show that?
    Consider this sequence <br />
A_n  = \left( {0,\frac{1}<br />
{n}} \right],\,n \in \mathbb{Z}^ +   .
    In what sense are you saying that sequence converges?
    It has an empty intersection.
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  3. #3
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    Quote Originally Posted by Plato View Post
    Consider this sequence <br />
A_n  = \left( {0,\frac{1}<br />
{n}} \right],\,n \in \mathbb{Z}^ +   .
    In what sense are you saying that sequence converges?
    It has an empty intersection.

    Sorry i forgot to mention that: \forall n \in \mathbb{N}
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  4. #4
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    Quote Originally Posted by hiddy View Post
    Sorry i forgot to mention that: \forall n \in \mathbb{N}
    That changes nothing.
    Please answer my first question.
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  5. #5
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    I still don't get what you mean, that sequence you posted gets smaller with every step and goes to 0 which is not part of the sets.

    Do you mean that similar to sequences of real numbers you net a monotonic and bounded seq?
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  6. #6
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    This what you first posted.
    Quote Originally Posted by hiddy View Post
    A sequence of sets is i monotone if A_n \subseteq A_{n+1} respectively A_n \supseteq A_{n+1} .
    Show that every monotonic sequence of sets converged and calulate the limes.
    Now I ask you: What does it mean to say that a sequence of sets converges?
    Here is another example: A_n  = \left( {1 - \frac{1}{n},1} \right),\,n \in \mathbb{Z}^ +   = \mathbb{N}
    Does that sequence of sets converge?
    They are nested. They are bounded.
    But in what sense can on say that they converge?
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  7. #7
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    I give up, not my day not my question

    The only thing left i can add is that in the last example they converge from below but that doesnt help me much.
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  8. #8
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    Quote Originally Posted by hiddy View Post
    I give up, not my day not my question
    The only thing left i can add is that in the last example they converge from below but that doesnt help me much.
    Are you working with Moore-Smith convergence?
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  9. #9
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    ahm actually no, because my question is from basic probability theory we did not talk about nets. I know that nets are generalisiations of sequences but there hast to be an easy proof that an isotone sequence of sets converges.
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