Results 1 to 9 of 9

Math Help - Can a sequence has more than one accumulation point?

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Can a sequence has more than one accumulation point?

    Let a seqence be in a metric space, can it has more than one accumulation point?

    My answer is no, because if it has more than one, then it would converge to both points, implies they are the same. Would that be the right start?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    How are you handling divergent sequences?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2008
    Posts
    42
    I actually need to know the same answer. I was thinking that a sequence could have more than one accumulation point. For example the sequence {(-1)^n}? I may be wrong though. If there is more than one accumulation point, how would I go about proving that?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,677
    Thanks
    1618
    Awards
    1
    Quote Originally Posted by EricaMae View Post
    I actually need to know the same answer. I was thinking that a sequence could have more than one accumulation point. For example the sequence {(-1)^n}? I may be wrong though. If there is more than one accumulation point, how would I go about proving that?
    You are correct.

    The sequence x_n  = \left( { - 1} \right)^n  + \frac{1}{n} has two accumulation points, \{1,-1\}
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    I found these pages in my textbook, I think it's what you're talking about....





    It starts on the first page and then it's the top of the second page.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Showcase_22 View Post
    I found these pages in my textbook, I think it's what you're talking about....





    It starts on the first page and then it's the top of the second page.
    An accumulation point of a squence is not the same thing as the limit of the sequence. An accumlation point may be taken to be defined as a limit of a sub-sequence, or alternativly as point such that every open interval containing the point also contains an element of the sequence.

    See Plato's post for an example.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    oops sorry.

    Could this adapted somehow to prove that a sequence can have more than one accumulation point?

    I am aware that the proof in the textbook proves a sequence cannot have more that one limit, so I am doubtful, but it would be great if it could provide a starting point.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Showcase_22 View Post
    oops sorry.

    Could this adapted somehow to prove that a sequence can have more than one accumulation point?

    I am aware that the proof in the textbook proves a sequence cannot have more that one limit, so I am doubtful, but it would be great if it could provide a starting point.
    Plato's example is sufficient to show this.

    RonL
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    R
    Posts
    481
    Thanks
    2
    If you deal with a bounded sequence which does not have limit.
    Then this means that this sequence have more than one convergent subsequences.
    Refer the example given by Plato (see also Bolzano?Weierstrass theorem - Wikipedia, the free encyclopedia).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Sequence and accumulation point
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: June 24th 2011, 10:02 AM
  2. accumulation point help
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: February 4th 2010, 11:30 AM
  3. Inf A= 0 <=> 0 is an accumulation point
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 7th 2009, 02:51 AM
  4. accumulation point
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 19th 2009, 02:11 PM
  5. Accumulation Point
    Posted in the Calculus Forum
    Replies: 7
    Last Post: August 19th 2007, 01:53 AM

Search Tags


/mathhelpforum @mathhelpforum