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Math Help - i cant understand this liminf/sup definition

  1. #1
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    i cant understand this liminf/sup definition

    i got this definition

    http://img515.imageshack.us/img515/5666/47016823jz1.gif

    what is the meaning of n>=0 under a sup.
    sup is not a limit
    its only a number
    we cant put index under it
    what is the meaning of this indexes?
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  2. #2
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    Quote Originally Posted by transgalactic View Post
    i got this definition

    http://img515.imageshack.us/img515/5666/47016823jz1.gif

    what is the meaning of n>=0 under a sup.
    sup is not a limit
    its only a number
    we cant put index under it
    what is the meaning of this indexes?
    Remember that if x_n is bounded then \limsup x_n = \lim \left( \sup \{ x_k | k\geq n\} \right).
    The sequence, \sup \{ x_k | k\geq n\} is non-increasing, therefore its limits is its infimum.
    Thus, \limsup x_n = \inf \{ \sup\{ x_k | k\geq n\} | n\geq 0 \}
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  3. #3
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    why you are saying that its not increasing
    <br /> <br />
\sup \{ x_k | k\geq n\}<br />

    its a supremum its only one number its the least upper bound of this sequence
    its a constant number it cannot increase or decrease.

    and then you are taking the great low bound of this constant number

    ???
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  4. #4
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    Quote Originally Posted by transgalactic View Post
    why you are saying that its not increasing
    <br /> <br />
    " alt="\sup \{ x_k | k\geq n\}
    " />

    its a supremum its only one number its the least upper bound of this sequence
    its a constant number it cannot increase or decrease.

    and then you are taking the great low bound of this constant number

    ???
    If \{ x_n \} is a bounded sequence which is non-increasing i.e. x_0 \geq x_1 \geq x_2 \geq x_3 ...
    Then  \lim x_n = \inf \{ x_0,x_1, x_2,... \}.
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    If \{ x_n \} is a bounded sequence which is non-increasing i.e. x_0 \geq x_1 \geq x_2 \geq x_3 ...
    Then  \lim x_n = \inf \{ x_0,x_1, x_2,... \}.
    ok i understood the last inf putting step.

    but why you are saying that
    \sup \{ x_k | k\geq n\}
    is not increasing.
    you are taking a bounded sequence and you get one number
    which is SUP (its least upper bound)
    thats it
    no more members

    ??
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