Results 1 to 2 of 2

Thread: When limit superior diverges

  1. #1
    Junior Member
    Joined
    Mar 2011
    Posts
    49

    When limit superior diverges

    Prove that $\displaystyle \lim\sup x_n=\infty\iff \forall M>0,\forall n\in\mathbb N,\exists k_0\ge n$ so that $\displaystyle x_k>M.$

    I think is an easy problem, but I'm confused, the statement establishes that $\displaystyle x_k$ is bounded below, but not above.

    How to prove this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by Connected View Post
    Prove that $\displaystyle \lim\sup x_n=\infty\iff \forall M>0,\forall n\in\mathbb N,\exists k_0\ge n$ so that $\displaystyle x_k>M.$

    I think is an easy problem, but I'm confused, the statement establishes that $\displaystyle x_k$ is bounded below, but not above.

    How to prove this?
    Which part are you having trouble with? Try firstly the only if statement. Then, you want to prove that $\displaystyle \limsup x_n=\infty$ but evidently by assumption you have that $\displaystyle \displaystyle \sup_{n\geqslant N}x_n\geqslant M$ for every $\displaystyle M\in\mathbb{R}^+$ so that $\displaystyle \displaystyle \limsup x_n=\lim_{N\to\infty}\sup_{n\geqslant N}x_n\geqslant M$ from where the conclusion follows since $\displaystyle M$ were arbitrary (intuitively you can take the limit as $\displaystyle M\to\infty$ of both sides of this last expression, but the left side is 'unaffected' by the limit since there is no $\displaystyle M$)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. limit superior of sequence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 13th 2012, 01:24 PM
  2. limit superior inequality
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 5th 2010, 11:47 AM
  3. limit superior ....
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 28th 2010, 02:57 PM
  4. Limit Superior
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Mar 7th 2010, 04:47 AM
  5. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Sep 3rd 2009, 05:05 PM

Search Tags


/mathhelpforum @mathhelpforum