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Question reagarding liminf definition

  1. Jan 9, 2009 #1
    regarding this definition

    [itex]

    a = \lim \bigg( \inf \{ a_k | k\geq n\} \bigg)
    [/itex]

    the sequence
    [itex]

    \inf \{ a_k | k\geq n \}
    [/itex]

    is non decreasing . its inf gets bigger or not changing in each following sequence
    so its limit is its least upper bound

    am i correct??
     
  2. jcsd
  3. Jan 9, 2009 #2

    HallsofIvy

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  4. Jan 10, 2009 #3
    what is the relations between this liminf and
    all the members in the sequence

    is it bigger or smaller then all of them?
     
  5. Jan 10, 2009 #4

    HallsofIvy

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    There is no requirement that a "liminf" be smaller or larger than all members of the sequence. The liminf of a sequence is the least upper bound of all subsequential limits.

    For example, suppose [itex]a_n[/itex] is (n-6)/2n if n is odd, -(n-6)/2n if n is even. Then {[itex]a_n[/itex]} is {-5/2, 1, -1/2, 1/4, -1/10, 0, 1/14, -1/8, ...}. For n odd, we have a sequence that converges to 1/2. For n even, we have a sequence that converges to -1/2. The liminf is the smaller of those, -1/2 but there is a number in the sequence less than -1/2. The limsup is 1/2 but there is a term of the sequence larger than 1/2. We can change any finite number of terms in a sequence with changing any subsequential limits so there cannot be any relation between the limit or liminf or limsup and individual terms of the sequence.
     
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