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Math Help - Cluster Points in a Sequence

  1. #1
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    Cluster Points in a Sequence

    Is there any sequence that has infinitely many cluster points?If there is,what are the sequences that have this property?
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  2. #2
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    Since \mathbb{Q} is countable, you can consider the sequence (q_n)_n formed by all the rationals, obtaining then a sequence such that its cluster points are ALL the real numbers!
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  3. #3
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    Quote Originally Posted by Enrique2 View Post
    Since \mathbb{Q} is countable, you can consider the sequence (q_n)_n formed by all the rationals, obtaining then a sequence such that its cluster points are ALL the real numbers!
    Hmm I don't quite understand.It would be helpful if you could give some concrete examples.

    Apart from the sequences formed by rational numbers,is there any sequence in \mathbb{R} that has this property as well?
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    It is a concrete example!!! \mathbb{Q} is countable and dense. Countable means that there exists a bijective aplication \pi:\mathbb{N}\to \mathbb{Q}. Hence (\pi(n))_n is a sequence containing ALL the rational numbers (not formed by rational numbers). Since \mathbb{Q} is dense in \mathbb{R}, every real point is a cluster point of this sequence.
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  5. #5
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    Quote Originally Posted by Enrique2 View Post
    It is a concrete example!!! \mathbb{Q} is countable and dense. Countable means that there exists a bijective aplication \pi:\mathbb{N}\to \mathbb{Q}. Hence (\pi(n))_n is a sequence containing ALL the rational numbers (not formed by rational numbers). Since \mathbb{Q} is dense in \mathbb{R}, every real point is a cluster point of this sequence.
    Okay,I get it.Thanks very much.
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  6. #6
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    If you want a simple constructive formula then you could take x_n = \sin(\ln n), which oscillates slowly between 1 and +1, so that every point of the interval [1,1] is a cluster point. That also provides an answer to the question in your other long-running thread Limit of a Sequence.
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