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Math Help - divergent sequences

  1. #1
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    divergent sequences

    Question:
    A sequence diverges if it has a subsequence which diverges.

    Is this statement true? Is there a counterexample?
    (Intuitively I would say this is false.)

    Thanks.
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  2. #2
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    Re: divergent sequences

    Quote Originally Posted by CountingPenguins View Post
    Question:
    A sequence diverges if it has a subsequence which diverges.
    Can you prove this: If a sequence converges the any of its subsequences converge to the same limit.
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  3. #3
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    Re: divergent sequences

    Yes, is the converse true?
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  4. #4
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    Re: divergent sequences

    Quote Originally Posted by CountingPenguins View Post
    Yes, is the converse true?
    Not the converse, the contrapositive is true.
    If a subsequence diverges then the sequence is divervent.
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  5. #5
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    Re: divergent sequences

    The question asks:
    True or False, if true prove, if false show a counterexample.

    A sequence s_n diverges if it has a subsequence which diverges.
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  6. #6
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    Re: divergent sequences

    Quote Originally Posted by CountingPenguins View Post
    The question asks:
    True or False, if true prove, if false show a counterexample.
    A sequence s_n diverges if it has a subsequence which diverges.
    If a sequence converges then any subsequence of that sequence also converges.

    The contrapositive which is also true is:
    If a sequence diverges then then any super-sequence of that sequence also diverges.
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  7. #7
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    Re: divergent sequences

    I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.

    I just keep thinking there is a counterexample to this.

    Is it a fact that if there is a divergent subsequence the whole sequence diverges?
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  8. #8
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    Re: divergent sequences

    Quote Originally Posted by CountingPenguins View Post
    I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.
    I just keep thinking there is a counterexample to this.
    Is it a fact that if there is a divergent subsequence the whole sequence diverges?
    Do you understand what the exact definition of sub-sequence means?

    If you do, how is the a counter-example to anything?
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  9. #9
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    Re: divergent sequences

    Quote Originally Posted by CountingPenguins View Post
    I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.
    I just keep thinking there is a counterexample to this.
    Is it a fact that if there is a divergent subsequence the whole sequence diverges?
    Do you understand what the exact definition of sub-sequence means?

    If you do, how is the a counter-example to anything?
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  10. #10
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    Re: divergent sequences

    Quote Originally Posted by CountingPenguins View Post
    I suppose I'm stuck because I'm reading this question as follows: If a sequence has a subsequence which diverges, then the sequence diverges.
    I just keep thinking there is a counterexample to this.
    Is it a fact that if there is a divergent subsequence the whole sequence diverges?
    Do you understand what the exact definition of sub-sequence means?

    If you do, how is this a counter-example to anything?
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  11. #11
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    Re: divergent sequences

    I finally see what you have been saying. Thanks.
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