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Math Help - About accumulation points..

  1. #1
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    About accumulation points..

    Hi,

    I had a problem proving this theorem:

    Let {sn} be a bounded sequence of real numbers. Assume S = {sn: n= 1, 2, ...} is infinite. Then {sn} is convergent if and only if S has exactly one accumulation point.

    But in another page of my notebook, it says:
    Accumulation points can be more than one or even infinitely many while the limit is unique.

    Are they opposite? Thanks for any help.
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  2. #2
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    Quote Originally Posted by truevein View Post
    I had a problem proving this theorem:
    Let {sn} be a bounded sequence of real numbers. Assume S = {sn: n= 1, 2, ...} is infinite. Then {sn} is convergent if and only if S has exactly one accumulation point. But in another page of my notebook, it says: Accumulation points can be more than one or even infinitely many while the limit is unique. Are they opposite?
    No, they are not opposite. A convergent sequence has exactly one accumulation point. A set can have many accumulation points.
    The set [0,1] has infinitely many accumulation points.
    The sequence \left(\frac{1}{n}\right),~n\in \mathbb{Z}^+ has exactly one.
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    Thanks! Then the problem is one being a sequence and other a set. I'll try it again.
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  4. #4
    Senior Member roninpro's Avatar
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    That doesn't quite resolve the issue. For example, the sequence a_n=(-1)^n(1-1/n) has two accumulation points: 1 and -1.
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    Quote Originally Posted by roninpro View Post
    That doesn't quite resolve the issue. For example, the sequence a_n=(-1)^n(1-1/n) has two accumulation points: 1 and -1.
    You missed the qualifying adjective, convergent .
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  6. #6
    Senior Member roninpro's Avatar
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    I was actually responding the post above mine. I thought that the poster did not understand that a sequence can also have multiple accumulation points. Maybe I misunderstood the spirit of the question. Either way, sorry for the confusion.
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    Quote Originally Posted by roninpro View Post
    I was actually responding the post above mine.
    This why we hope that you will use the "reply with quote" option.
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