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Definition of liminf of sequence of functions?

  1. May 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi I've come across the term lim inf ##f_n## in my text but am not sure what it means.

    ##\lim \inf f_n = \sup _n \inf _{k \geq n} f_k##

    In fact, I am not sure what is supposed to be the output of lim inf f? That is, is it supposed to return a real-valued number, or a function itself?

    Generally for a real-valued function ##\inf f## refers to ##\inf_x f(x)##. That is, it returns the largest real-valued number smaller than f(x) for all x. If that's the case, then it should follow that ##\lim \inf f_n## also returns a real-valued number? But I've always thought the implication of lim inf and lim sup is that if ##f_n## converges uniformly to ##f## then

    ##\lim \inf f_n = \lim \sup f_n = \lim f_n = f##

    but that doesnt seem to hold if i use this definition? Any help or clarification would be greatly appreciated. Thanks!
     
  2. jcsd
  3. May 3, 2013 #2
    I really like this image, from Wikipedia. It helped get me started in understanding intuitively what limsup and liminf mean. Hopefully it does the same for you.
     
  4. May 3, 2013 #3
    Hi Mandelbroth,

    Thanks for your reply! However, that picture only describes the limsup/liminf of a sequence of points, which is itself a point, which is easier to intuit for me. But what I'm wondering is what is the limsup/liminf of a sequence of functions supposed to be? Is it to be a function itself? Or just a point? I am unclear as to the definition itself of limsup for sequences of functions....

    So far I'm thinking it shouldnt be intuitive for limsup/liminf f_n to be a point, since as I mentioned earlier if ##f_n## converges to ##f## uniformly it ought to be the case that ##\lim \sup f_n = \lim \inf f_n = \lim f_n##? Thanks again for any help
     
  5. May 3, 2013 #4

    Dick

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    Yes, it's a function. The value of lim inf ##f_n## at a point x is the lim inf of sequence of numbers ##f_n(x)## as n->infinity.
     
  6. May 4, 2013 #5

    micromass

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    If you understand limsup/liminf of sequences of points, then this isn't too hard. Basically, you take a sequence of functions ##(f_n)_n##. Now, if I take a fixed ##x##, then ##x_n = f_n(x)## is a sequence of points. So the liminf makes sense. Now, we define

    [tex]f(x) = \liminf x_n[/tex]

    And we do that for any point. So the liminf of a sequence of functions is again a function ##f## which satisfies that

    [tex]f(x) = \liminf f_n(x)[/tex]

    for any ##x##. So you evaluate the liminf pointswize.

    This is true. But uniform convergence isn't even needed. Pointswize convergence is enough.
     
  7. May 4, 2013 #6
    Thanks guys! That makes perfect sense!
     
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