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Math Help - limits superior and inferior

  1. #1
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    limits superior and inferior

    suppose
    \lim sup_{n\to\infty}A_n=\bigcap_{n=1}^{\infty}\bigcup_  {m\ge n}A_m
    and
    \lim inf_{n\to\infty}A_n=\bigcup_{n=1}^{\infty}\bigcap_  {m\ge n}A_m

    where \bigcup_{m\ge n}A_m=A_n\cup A_{n+1}\cup .... and \bigcap_{m\ge n}A_m=A_n\cap A_{n+1}\cap ....

    Then given that \lim inf_{n\to\infty}A_n\in \mathcal{A} and \lim sup_{n\to\infty}A_n\in \mathcal{A}

    If A_n\rightarrow A, i.e. \lim_{n\to\infty}1_{A_n}=1_A, so 1_{\text{lim sup}_{n\to\infty}A_n}=\lim sup_{n\to\infty}1_{A_n}=1_A=\lim inf_{n\to\infty}1_{A_n}=1_{\text{lim inf}_{n\to\infty}A_n}

    Then how do I show that \lim sup_{n\to\infty}1_{A_n}-\lim inf_{n\to\infty}1_{A_n}=1_{(\text{lim sup}_nA_n\setminus \text{lim inf}_nA_n)}

    where A\setminus B=A\cap B^c \text{ whenever } B\subset A.

    I was wondering since \lim sup_{n\to\infty}1_{A_n} \text{ and } \lim inf_{n\to\infty}1_{A_n} are the same, why wouldn't the end result goes to 0?
    Last edited by noob mathematician; September 11th 2010 at 09:47 AM.
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  2. #2
    Moo
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    Hello,

    There's no element telling you that limsup and liminf are the same. The equality you want to prove isn't dependent on the fact that A_n\to A.

    As to prove the equality, when does 1_{(\limsup_n A_n\setminus \liminf_n A_n)} equal 1 and equal 0 ? What does it imply for the LHS ? It's not that hard.
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  3. #3
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    Quote Originally Posted by Moo View Post
    Hello,

    There's no element telling you that limsup and liminf are the same. The equality you want to prove isn't dependent on the fact that A_n\to A.

    As to prove the equality, when does 1_{(\limsup_n A_n\setminus \liminf_n A_n)} equal 1 and equal 0 ? What does it imply for the LHS ? It's not that hard.
    Ok. It is obvious that: \lim inf_{n\to\infty}A_n\subset \lim sup_{n\to\infty}A_n

    So when \lim sup_{n\to\infty}1_{A_n}-\lim inf_{n\to\infty}1_{A_n}, I will get A \cap B^c letting  A= \lim sup_{n\to\infty}1_{A_n} \text{ and } B= \lim inf_{n\to\infty}1_{A_n}.

    Is this the correct interpretation?
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