Results 1 to 12 of 12

Math Help - limit superior/inferior proof

  1. #1
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53

    limit superior/inferior proof

    *x_n indicates x sub n
    *>= indicates greater than or equal to

    Let (x_n) be a bounded sequence and for every n in natural numbers, let s_n:= sup {x_k : k>=n} and t_n:= inf {x_k : k>=n}

    Prove that (s_n) and (t_n) are monotone and convergent. Also prove that if lim(s_n) = lim(t_n), then (x_n) is convergent.

    Can someone get me started in the right direction? Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    You want to show that s_n is decreasing and t_n is increasing.

    These should be fairly obvious since \{ x_k|k\geq n+1\}\subseteq \{ x_k|k\geq n\}.

    Note that "convergent" allows for the possibility of converging to \infty or -\infty.

    I assume that we're working in a complete space (maybe the reals or complexes?) If so, then you only need show that the sequences are Cauchy sequences.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53
    where do you get that the {x_k : k>= n +1} is a subset of {x_k : k>= n} ?? And we haven't gotten to Cauchy sequences yet
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by mremwo View Post
    where do you get that the {x_k : k>= n +1} is a subset of {x_k : k>= n} ??
    That is simply saying \{x_6,x_7,x_8,\cdots\}\subset\{x_5,x_6,x_7,\cdots\  } for example.

    Do you understand that if A\subset B and U=\sup(B) then \sup(A)\le U~?
    Last edited by Plato; February 9th 2011 at 02:53 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    Quote Originally Posted by Plato View Post
    That is simply saying \{x_5,x_6,x_7,\cdots\}\subset\{x_6,x_7,x_8,\cdots\  } for example.
    [/tex]
    You got your sets backwards. It should be \{x_6,x_7,x_8,\cdots\}\subset\{x_5,x_6,x_7,\cdots\  } .
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Please see my edit. Thanks
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    I assume that we're working with real numbers here (you haven't clarified this).

    Do you already have the theorem that an increasing sequence that is bounded above converges? (and similarly for decreasing and bounded below)
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53
    Yes, I'm working with real numbers, sorry.
    I do have that theorem, but how would I go about proving that each of the sequences converge?
    I did the proving they are monotone part (thanks), now I'm stuck at proving they converge.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    Well once you prove monotone you're essentially done. For example, if you show that the sequence is increasing, then there are 2 cases.

    Case 1 - the sequence is bounded above. Then the sequence converges to a finite number.
    Case 2 - the sequence is not bounded above. Then the sequence converges to infinity.

    Edit: I just noticed that you are given that original sequence is bounded. Thus the sup and inf of the sequence exist. So the other 2 sequence are bounded as well. So Case 1 only applies.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53
    Thank you very much, I think I'm starting to get it.

    One question: I am a little bit confused about the relationship the x_n sequence has with s_n and t_n. I mean I know that s_n is the sup the set that was given and t_n the inf, but I don't get what the k's have to do with it. Also, it then asks me to show if lim(s_n)=lim(t_n) then x_n is convergent. Why is that?

    EDIT: As in, how does (x_n) being bounded tell you that the sup and inf of the other sequences are bounded as well?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by mremwo View Post
    EDIT: As in, how does (x_n) being bounded tell you that the sup and inf of the other sequences are bounded as well?
    If the sequence x_n is bounded then  \left( {\exists B > 0} \right)\left( {\forall n} \right)\left[ { - B \leqslant x_n  \leqslant B} \right].
    That insures the sequence has a supremum and an infimum.
    So it is true of any of sub-sequence of the sequence.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53
    Okay, I got that now. Now how would lim(s_n)=lim(t_n) show that (x_n) is convergent?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Limit Superior and Limit Inferior
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 11th 2011, 04:43 AM
  2. limit superior and limit inferior
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 17th 2010, 07:28 PM
  3. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 3rd 2009, 05:05 PM
  4. Find the superior and inferior limit??
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 23rd 2009, 02:57 PM
  5. Replies: 2
    Last Post: October 4th 2008, 04:57 PM

Search Tags


/mathhelpforum @mathhelpforum