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Math Help - Cluster point:

  1. #1
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    Cluster point:

    Q: Find a sequence having only one cluster point, yet not convergent.

    Solution:
    Let N = 1,2,1,3,1,4,1,5,....,1,n,.....

    Clearly 1 is a cluster points of N, yet N is unbounded.


    Just wondering if this solution is correct? Any suggestions!!!
    Thanx
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  2. #2
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    Quote Originally Posted by Sheikh_fiji View Post
    Q: Find a sequence having only one cluster point, yet not convergent.

    Solution:
    Let N = 1,2,1,3,1,4,1,5,....,1,n,.....

    Clearly 1 is a cluster points of N, yet N is unbounded.


    Just wondering if this solution is correct? Any suggestions!!!
    Thanx
    For a sequence \{a_n\} be say that a is an accumulation point if for any \epsilon > 0 there exists a distinct a_N so that |a_N-a| < \epsilon. In your example 1 is not an accumulation point because if \epsilon = \tfrac{1}{2} the only terms of the sequence that are within \epsilon to 1 is just 1 itself. It needs to be distinct from 1.
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  3. #3
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    ...but easily corrected by replacing 1 with any convergent sequence?
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