LimSup and LimInf

Discussion in 'Undergraduate Math' started by khosaa, Feb 21, 2011.

  1. khosaa

    khosaa Guest

    Hi,

    It seems that there are two common definitions for the terms "Limit
    Superior" and "Limit Inferior": one pertains to sequences and the
    other to sets.

    I am curious if the two defns are in any way related. I can't seem to
    find such a connection, but perhaps there is some clever way to cast
    the LimSup for a sequence in terms of sets that I'm somehow missing.

    Thanks,
    Fran
     
    khosaa, Feb 21, 2011
    #1
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  2. Given a sequence of real numbers {a_n}, let A_n = (-oo,a_n).

    The Limsup A_n = (-oo,limsup a_n), where Limsup is the limit superior
    of sets, and limsup is the limit superior of sequences. Likewise,
    Liminf A_n = (-oo,liminf a_n).
     
    Arturo Magidin, Feb 21, 2011
    #2
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  3. I'm not sure that's exactly correct. Suppose a_n = 1/n, A_n = (-oo, 1/
    n); then limsup a_n = lim a_n = 0, while Limsup A_n = Lim A_n = (-oo,
    0], not (-oo, 0).

    As an alternative answer to the OP, if f_n(x) is the characteristic
    function of A_n, I believe that limsup f_n(x) is the characteristic
    function of Limsup A_n, and liminf f_n(x) is the characteristic
    function of Liminf A_n.
     
    Butch Malahide, Feb 21, 2011
    #3
  4. Right; thanks for the correction. Endpoints may or may not be
    included; Limsup A_n is of the form (-oo,k) or (-oo,k], and k equals
    limsup a_n; and liminf A_n is of the form (-oo,m) or (-oo,m], and m
    equals liminf a_n.
     
    Arturo Magidin, Feb 21, 2011
    #4
  5. I can imagine at least four different things you might mean by that.
    If you say exactly what two definitions you're referring to you're
    more likely to get an answer to the question you meant to ask.
     
    David C. Ullrich, Feb 21, 2011
    #5
  6. khosaa

    khosaa Guest


    Hi David,

    I'll simply state the defns for LimSup that I had in mind (since the
    corresponding defns for LimInf should be obvious)

    The defn I had in mind for the sequence (a_n) was as follows:
    then LimSup (a_n) = Inf (over all natural numbers m) { Sup{a_m,a_m
    +1,...}}

    The defn I had in mind for the family of sets {A_n} was
    LimSup A_n = Intersection for m=1 to +Infinity [ Union for n >= m of
    A_n]

    Butch and Arturo gave a nice explanation of how the two are related:
    For any natural number n, define b_n = Sup{a_n,a_n+1,...}
    then we have a_m <= b_n for every m>= n implies
    (-00,a_m) subset (-00, b_n) for every m>= n implies
    Union from m=n to +infinity (-00,a_m) subset (-00,b_n)
    but inf of all b_n can be obtained simply by taking the intersection
    of all the (-00,b_n)

    I know of only these two defns (which I found in the Harper Collins
    dictionary of math). Curious what the others are.

    Fran
     
    khosaa, Feb 23, 2011
    #6
  7. khosaa

    khosaa Guest

    Hi Arturo,

    Thank you very much for the nice explanation. Definitely would not
    have seen it on my own. ;>)

    Fran
     
    khosaa, Feb 23, 2011
    #7
  8. Fine, then - that's more or less what Butch and Arturo thought
    you were talking about, so never mind my curiosity about
    what you meant.

    (In case you're wondering what I was wondering about: there
    are various ways of defining the llim sup of a _sequence_
    that refer to various _sets_, for example lim sup a_n is
    the largest element of S, where S is the set of all accumulation
    points. Or, lim sup a_n = L if and only if the set
     
    David C. Ullrich, Feb 23, 2011
    #8
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