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Math Help - Limit superior and limit inferior

  1. #1
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    Limit superior and limit inferior

    I get the definition of limit superior and limit inferior, but I can't start a proof for these properties:

    \underline{\lim }{\,{x}_{n}}=-\overline{\lim }\left(-{{x}_{n}}\right)

    \underline{\lim }{\,{x}_{n}}\le\overline{\lim }{\,{x}_{n}}

    \underline{\lim }{\,{x}_{n}}+\underline{\lim  }{\,{y}_{n}}\le\underline{\lim  }\left({{x}_{n}}+{{y}_{n}}\right)\le\overline{\lim  }{\,{x}_{n}}+\underline{\lim }{\,{y}_{n}}\le\overline{\lim  }\left({{x}_{n}}+{{y}_{n}}\right)\le\overline{\lim  }{\,{x}_{n}}+\overline{\lim }{\,{y}_{n}}.

    Hope you may enlight me.
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  2. #2
    Super Member girdav's Avatar
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    Did you try to use the properties of \inf and \sup ?
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  3. #3
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    Oh, not really.

    How would you prove the third one? Can you show me?

    Thanks!
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  4. #4
    Super Member girdav's Avatar
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    Use \displaystyle\inf_{n\geq k}(a_n+b_n)\geq \inf_{n\geq k}a_n+\inf_{n\geq k}b_n for the first inequality. For the second, you have to show that \liminf (x_n+y_n)-\limsup x_n\leq \liminf y_n. But \liminf (x_n+y_n)-\limsup x_n =\liminf (x_n+y_n)+\liminf (-x_n)\leq \liminf (x_n+y_n-x_n).
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  5. #5
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    So for the first one I use \sup (-x_n)=-\inf x_n, by taking limits I get the result. Is it correct?

    The second one follows easily from \inf x_n\le \sup x_n, right?
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  6. #6
    Super Member girdav's Avatar
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    You have to tell where you take the \sup. The definition is \displaystyle \limsup_n x_n = \inf_{k\in\mathbb N}\sup_{n\geq k} x_n.
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  7. #7
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    Oh well, so I need to add that to what I wrote.

    But by adding this, is it correct? If not, how to do it?
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