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Math Help - lim inf/sup innequality question...

  1. #1
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    lim inf/sup innequality question...

    x_n and y_n are bounded

    \limsup x_n + \limsup y_n \geq \lim x_{r_n} + \lim y_{r_n}

    this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
    the sum of limits is the limit of sums so we get one limit
    and the of two convergent sequence is one convergent sequence
    \lim (x_{r_n}  + y_{r_n}) = \limsup (x_n+y_n)

    this is true because the sequence is constructed from a convergent to the sup sub sequences
    so they equal the lim sup of x_n+y_n

    now i need to prove that \limsup (x_n+y_n) \Rightarrow \liminf x_n + \limsup y_n

    i tried: <br />
\limsup (x_n+y_n) = \limsup x_n + \limsup y_n \Rightarrow \liminf x_n + \limsup y_n<br />
    lim sup i always bigger then lim inf

    so its true

    did i solved it correctly?
    Last edited by mr fantastic; March 11th 2009 at 11:06 PM. Reason: Replaced tex tags with the correct latex tags (for MHF, anyway ....) and tidied up the latex a bit
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by transgalactic View Post
    x_n and y_n are bounded
    <br />
lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n<br /> <br />
    this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
    the sum of limits is the limit of sums so we get one limit
    and the of two convergent sequence is one convergent sequence
    <br />
lim(x_r_n+y_r_n)=limsup(x_n+y_n)<br />
    this is true because the sequence is constructed from a convergent to the sup sub sequences
    so they equal the lim sup of x_n+y_n

    now i need to prove that
    <br />
limsup(x_n+y_n)=>liminf x_n+limsup y_n<br />

    i tried:
    <br />
limsup(x_n+y_n)=limsup x_n+limsup y_n=>liminf x_n+limsup y_n<br />
    lim sup i always bigger then lim inf

    so its true

    did i solved it correctly?

    I'm not sure what you want to prove.
    Is r_n a particular subsequence?
    And why do we know that there is a limit to x_{r_n}
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  3. #3
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    x_r_n is a convergent subsequence of x_n


    y_r_n is a convergent subsequence of y_n

    r_n is just a sign to demonstrate that its a subsequence

    by weirshrass laws to any bounded sequence there is a convergent subsequence

    my proof is ok?
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  4. #4
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by transgalactic View Post
    x_n and y_n are bounded

    \limsup x_n + \limsup y_n \geq \lim x_{r_n} + \lim y_{r_n}

    this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
    the sum of limits is the limit of sums so we get one limit
    and the of two convergent sequence is one convergent sequence
    \lim (x_{r_n} + y_{r_n}) = \limsup (x_n+y_n)

    this is true because the sequence is constructed from a convergent to the sup sub sequences
    so they equal the lim sup of x_n+y_n

    now i need to prove that \limsup (x_n+y_n) \Rightarrow \liminf x_n + \limsup y_n

    i tried: <br />
\limsup (x_n+y_n) = \limsup x_n + \limsup y_n \Rightarrow \liminf x_n + \limsup y_n<br />
    lim sup i always bigger then lim inf

    so its true

    did i solved it correctly?

    I'm still having trouble figuring out what you want to prove.
    Also I don't think that
    \limsup (x_n+y_n) = \limsup x_n + \limsup y_n.
    What if x_n=0,1,0,1,0,1,...
    and y_n=1,0,1,0,1,0,..., then
    x_n+y_n=1,1,1,1,1,1,..., which has a limit and limit soupy of 1.
    But the \limsup x_n + \limsup y_n=1+1=2.

    If you just want to prove
    \limsup x_n + \limsup y_n \geq \lim x_{r_n} + \lim y_{r_n}
    well \limsup x_n \geq \lim x_{r_n}
    and \limsup y_n \geq \lim y_{r_n}.
    Now just add, but I'm still lost as to what you want.
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  5. #5
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    <br /> <br />
\lim (x_{r_n}  + y_{r_n}) = \limsup (x_n+y_n)<br />

    why its correct
    what do i need to say so it will be valid
    ??
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  6. #6
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by transgalactic View Post
    <br /> <br />
\lim (x_{r_n} + y_{r_n}) = \limsup (x_n+y_n)<br />

    why its correct
    what do i need to say so it will be valid
    ??

    Let w_n=x_n+y_n.
    Now I still need to understand this sequence r_n.
    Is r_n the sequence so that
    \limsup w_n=\lim w_{r_n}?
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  7. #7
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    r_n is not a sequence its just a way for me so sign a subsequence .

    x_n is a bounded sequence with index n
    and i want a subsequence to x_n
    so i call it x_{r_n} where n is the index again

    its just a way to sign a subsequence

    is it ok now?
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  8. #8
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    can you show an example like this

    there is such things on the internet
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