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Math Help - Unique Cluster Point => Convergent?

  1. #1
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    Unique Cluster Point => Convergent?

    The question is:

    Let the sequence  \{ x_n \} be in  \mathbb {R} that has a unique cluster point, does that implies the sequence converges?

    I was looking up the theorems, and I believe it is false since we did not say that the sequence is bounded. But are there any examples for that?

    Thanks.
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  2. #2
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    Consider the sequence t_n  = \left\{ {\begin{array}{ll}<br />
   n & {\mbox{n odd}}  \\    {\frac{1}{n}} & {\mbox{n even}}  \\ \end{array} } \right.
    Does it have a unique cluster point?
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  3. #3
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    Yes, 0 is the unique cluster point, since for any natural N, I can find all even number of n greater than or equal N such that | t_n - 0 | < \epsilon .

    And this sequence doesn't converge, since if I let  \epsilon = 1/2 , let  N \in \mathbb {N} , then whenever  n \geq N , I would have  |t_n| > 1/2 when n is odd.

    Is my argument proper? Thank you.
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  4. #4
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    Quote Originally Posted by tttcomrader View Post
    Is my argument proper?
    Yes it works.
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