Results 1 to 3 of 3
Like Tree1Thanks
  • 1 Post By jakncoke

Math Help - Understanding Limit Supremum and Limit Infimum

  1. #1
    Member
    Joined
    Oct 2010
    Posts
    150

    Understanding Limit Supremum and Limit Infimum

    It's easy for me to say that the Limit Supremum is the "largest subsequential limit" and the Limit Infimum is the "smallest subsequential" limit, but that's about as far as my understanding goes.

    What's a good, intuitive way to understand these concepts?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2010
    Posts
    150

    Re: Understanding Limit Supremum and Limit Infimum

    Any help on this? I studied some more today but it's still a little fuzzy :/
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member jakncoke's Avatar
    Joined
    May 2010
    Posts
    387
    Thanks
    80

    Re: Understanding Limit Supremum and Limit Infimum

    Its most easily understood with an example

    take the sequence S_n = \{5,6,7,2,4,-1,1,-1,1,-1,1,...\}
    Then the sup of S_n is the least upper bound, which would be 7. Basically you look at the entire S_n and say 7 is the smallest number which is an upper bound for this set.

    But for lim sup, you basically take a look at the tails of this sequence and see what the least upper bound is.

    By tail i mean S_n = \{5,6,7,2,4,-1,1,-1,1,-1,1,...\} is a tail
    S_n = \{6,7,2,4,-1,1,-1,1,-1,1,...\} is a tail
    S_n = \{7,2,4,-1,1,-1,1,-1,1,...\} is a tail
    S_n = \{2,4,-1,1,-1,1,-1,1,...\} is a tail

    im essentially removing the first element of the sequence, the second element, and third, and fourth etc... and taking the sup of those sequences.
    I do this an infinite amount of times.

    So for our sequence S_n the lim sup is 1.
    The sup of the first tail is 7
    sup of second tail is 7
    third is 7
    fourth is 4
    fifth is 4
    sixth is 1
    seventh is 1
    etc...

    Basically if you have a finite number of "big" numbers in the sequence, then the sup of the sequence only gives a distorted "least upper bound".
    but lim sup gives you the sup if you throw away the finite number of "big" numbers in the sequence

    in the sequence P_n = \{5,1,1,1,1,1,1,1,1,1,...\}
    sup of P_n is 5
    but lim sup of P_n is 1

    These same analogies can be extended for lim inf and inf.
    Thanks from divinelogos
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. limit and supremum
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 3rd 2012, 05:00 PM
  2. Supremum and infimum
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 19th 2010, 04:16 AM
  3. Using supremum and infimum
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 16th 2010, 11:34 AM
  4. Infimum and supremum
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: February 9th 2010, 04:57 PM
  5. supremum and infimum of a set
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: January 14th 2008, 01:26 PM

Search Tags


/mathhelpforum @mathhelpforum