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Math Help - limit

  1. #1
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    limit

    how do you prove that a sequence a_n has a limit iff lim sup a_n = lim inf a_n?
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  2. #2
    Senior Member Tinyboss's Avatar
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    Hint: You can find a subsequence that converges to the lim sup.
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  3. #3
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    Can i say that there exist a subseq f(x_nk) converges to say f(c) as k tends to infinity, where f(c) is lim sup then f(c) is also lim inf?

    Im still abit confused..
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  4. #4
    Senior Member Tinyboss's Avatar
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    Every sequence has a subsequence converging to the lim sup of the original sequence, and a subsequence converging to the lim inf of the original sequence. What do you know about subsequences of convergent sequences?
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  5. #5
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    They converge to the limit of the seq which is unique?
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  6. #6
    Senior Member Tinyboss's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    They converge to the limit of the seq which is unique?
    Yes, and that gives you the "only if" direction.
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  7. #7
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    And to prove the other direction, since limit exist, all subseq have to converge to the same value hence lim sup is equal to lim inf?
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  8. #8
    Senior Member Tinyboss's Avatar
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    No, that's the direction we just discussed. Now you need to show that lim sup = lim inf implies convergence.
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  9. #9
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    How do i show that?

    It seems direct to me that since the lim sup is lim inf, then all sunseq converge to the same value. Thus unique accumulation point and hence the seq converges.. Would an explaination like this work?
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  10. #10
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    i was thinking about it still and still cant figure out the other direction. could you please teach me how to proof?
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  11. #11
    Senior Member Tinyboss's Avatar
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    For the other direction, write down the definitions, formally in terms of epsilons and deltas, for limit, lim sup, and lim inf, then just crank it out. I'm not going to give a full solution for both directions.
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