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Math Help - Real analysis question on subsequence

  1. #1
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    Real analysis question on subsequence

    Hi, my question is:

    Let L be a real number and let (a_n) be a sequence of real numbers that does not converge to L (that is, it is either divergent or its limit is not equal to L). Use the definition of convergence to L to show that for some e greater than 0, (a_n) has a subsequence (a_n(_k)) such that (a_n(_k)) isn't in the interval (L - e, L+e) for all natural numbers k.

    Any help would be hugely appreciated
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  2. #2
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    Re: Real analysis question on subsequence

    Quote Originally Posted by sakuraxkisu View Post
    Let L be a real number and let (a_n) be a sequence of real numbers that does not converge to L (that is, it is either divergent or its limit is not equal to L). Use the definition of convergence to L to show that for some e greater than 0, (a_n) has a subsequence (a_n(_k)) such that (a_n(_k)) isn't in the interval (L - e, L+e) for all natural numbers k.

    To say that \left( {a_n } \right)\not \to L means that
    \left( {\exists \varepsilon  > 0} \right)\left( {\forall N} \right)\left( {\exists n_N  > N} \right)\left[ {\left| {L - a_{N_n } } \right| \geqslant \varepsilon } \right].

    Find the first N_1 then apply the above and find N_2>N_1 that works.

    By induction, find a sequence of integers N_1<N_2<\cdots<N_m<\cdots.
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  3. #3
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    Re: Real analysis question on subsequence

    Plato, great explanation, solved my other problem in the series also. I am glad that I bumped into this thread.
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  4. #4
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    Re: Real analysis question on subsequence

    Quote Originally Posted by SeirraFalcom View Post
    Plato, great explanation, solved my other problem in the series also. I am glad that I bumped into this thread.
    Update: I am stuck again in the test prep of my discrete math subject.
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