Results 1 to 4 of 4

Thread: lim inf contained in lim sup

  1. #1
    Junior Member TheProphet's Avatar
    Joined
    Aug 2011
    Posts
    29

    lim inf contained in lim sup

    Hey,

    $\displaystyle (A_{n})_{n \geq 1} $ a sequence of subsets of $\displaystyle \mathcal{A} $, a $\displaystyle \sigma $-field on $\displaystyle \Omega $.

    With
    $\displaystyle \lim_{n \to \infty} \inf A_{n} = \bigcup_{n=1}^{\infty} \bigcap_{m\geq n} A_{m} $
    and
    $\displaystyle \lim_{n \to \infty} \sup A_{n} = \bigcap_{n=1}^{\infty}\bigcup_{m \geq n} A_{m} $

    show that $\displaystyle \lim_{n \to \infty} \inf A_{n} \subset \lim_{n \to \infty} \sup A_{n} $

    Hints on how to start?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: lim inf contained in lim sup

    Let $\displaystyle x\in\liminf_{n\to +\infty} A_n$. We can find $\displaystyle n_0$ such that for $\displaystyle n\geq n_0$, $\displaystyle x\in A_n$. Now, you have to show that for all $\displaystyle k\in\mathbb{N}$, $\displaystyle x\in\bigcup_{j\geq k}A_j$.
    "To be in $\displaystyle \liminf_{n\to +\infty} A_n$" means "to be in all $\displaystyle A_n$ for $\displaystyle n$ large enough" whereas "To be in $\displaystyle \limsup_{n\to +\infty} A_n$" means "to be in infinitely many $\displaystyle A_n$".
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member TheProphet's Avatar
    Joined
    Aug 2011
    Posts
    29

    Re: lim inf contained in lim sup

    $\displaystyle x \in \lim_{n \to \infty} \inf A_{n} \Rightarrow \exists n_0 : x \in A_{n} \forall n \geq n_{0} $

    If $\displaystyle x \in \bigcup_{j \geq k} A_{j} $ but $\displaystyle x \notin \bigcup_{j \geq k+1} A_{j} $ this means $\displaystyle x \notin \lim_{n \to \infty} \inf A_{n} $ which is a contradiction.

    And clearly $\displaystyle x \in \bigcup_{j \geq 1} A_{j} $, whence

    $\displaystyle x \in \lim_{n \to \infty} \sup A_{n} $

    $\displaystyle \lim_{n \to \infty} \inf A_{n} \subset \lim_{n \to \infty} \sup A_n{n} $

    Is this ok?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: lim inf contained in lim sup

    Yes, it works.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subset contained
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Jan 31st 2011, 05:48 AM
  2. Percent contained
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Nov 10th 2009, 12:33 PM
  3. Two balls such that the bigger one is contained inside the smaller
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Sep 20th 2009, 02:16 PM
  4. Replies: 3
    Last Post: Feb 6th 2009, 08:37 AM
  5. Interval contained in most intervals
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Jul 7th 2008, 04:58 AM

Search Tags


/mathhelpforum @mathhelpforum