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Math Help - upper/lower sequence question..

  1. #1
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    upper/lower sequence question..

    if my sequence is

    1/2,2/3,3/4,4/5 ..

    why the upper sequence is 1

    and lower sequence is 1 too ??

    1 is not ever a member of a sequence.

    i got the idea that upper sequence is a sub sequence constructed from the highest members
    the closest to the upper bound

    1 is not even in the sequence??
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  2. #2
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    Quote Originally Posted by transgalactic View Post
    if my sequence is

    1/2,2/3,3/4,4/5 ..

    why the upper sequence is 1

    and lower sequence is 1 too ??
    I am assuming this question is about limsup's and liminf's.
    Let x_n = \tfrac{n}{n+1}.

    If a_n is the superior (upper) sequence then a_n = \sup \{ x_k | k\geq n \} = \sup \left\{ \tfrac{k}{k+1} \bigg| k\geq n \right\} = 1

    If b_n is the inferior (lower) sequence then b_n = \inf \{ x_k|k\geq n\} = \inf \left\{ \tfrac{k}{k+1} \bigg| k\geq n \right\} = \frac{n}{n+1}

    Therefore, \limsup x_n = \lim a_n = 1 and \liminf x_n = \lim a_n = 1.
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