1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I cant understand this explanation of limsup

  1. Jan 8, 2009 #1
    regarding this defintion


    i was told that
    Remember that if [itex]x_n[/itex] is bounded then [itex]\limsup x_n = \lim \left( \sup \{ x_k | k\geq n\} \right)[/itex].
    The sequence, [itex]\sup \{ x_k | k\geq n\}[/itex] is non-increasing, therefore its limits is its infimum.
    Thus, [itex]\limsup x_n = \inf \{ \sup\{ x_k | k\geq n\} | n\geq 0 \} [/itex][/quote]

    i cant understand the first part

    why he is saying that
    [itex]\sup \{ x_k | k\geq n\}[/itex]
    is not increasing.
    you are taking a bounded sequence and you get one number
    which is SUP (its least upper bound)
    thats it.
    no more members

  2. jcsd
  3. Jan 8, 2009 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    But [itex]\left{x_k|k\ge n}[/itex] is not a single sequence- it is a different sequence for every different n.

    For example, if [itex]{x_n= (-1)^n/n}= {-1, 1/2, -1/3, 1/4, -1/5, ...} then
    [itex]sup{x_k|k\ge 1}[/itex] is the largest of {-1, 1/2, -1/3, 1/4, -1/5, ...} which is 1/2. [itex]sup{x_k|k\ge 2}[/itex] is the largest of {1/2, -1/3, 1/4, -1/5, ...}, again 1/2. [itex]sup{x_k|k\ge 3}[/itex] is the largest of {-1/3, 1/4, -1/5, ...}, which is 1/4. Similarly, [itex]sup{x_k|k\ge 4}[/itex] is also 1/4 but [itex]sup{x_k|k\ge 5}[/itex] is 1/6, etc.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: I cant understand this explanation of limsup