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Lim inf = lim sup = lim {An}

  1. Oct 30, 2012 #1
    Hi,

    What is the proof that if series {An} is bound and its lim inf = lim sup = L, then lim A must be equal to L?
     
  2. jcsd
  3. Oct 30, 2012 #2

    HallsofIvy

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    What definition of "lim inf" and "lim sup" are you using?
     
  4. Oct 30, 2012 #3
    Lim inf = lowest partial limit, i.e. lowest limit amongst all the limits of all the sub-series.
    Lim sup = highest partial limit, i.e. highest partial limit amongst all the limits of all sub-series
     
  5. Oct 30, 2012 #4

    HallsofIvy

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    So you are saying that if a is a limit of any subsequence, then [itex]A\le a\le A[/itex]! What does that tell you?
     
  6. Oct 31, 2012 #5
    a is not the limit of ANY subsequence, but the smallest of the limits of the subsequences and the largest of the limits of the subsequences.
    How may I prove that the sequence itself converges to the same limit?
     
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