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Math Help - Sequence Question

  1. #1
    Newbie FishStyx's Avatar
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    Sequence Question

    Prove the following:

    If \{b_n\}_{n=1}^{\infty} converges to B \neq 0 and b_n \neq 0 for all n, then there is an M > 0 such that |b_n| \geq M for all n.

    Now, this sounds very similar to a Lemma that was proven in class.

    If \{b_n\}_{n=1}^{\infty} converges to B and B \neq 0, then there is a positive real number M and a positive integer N such that, if n \geq N, then |b_n| \geq M.

    I'm just having trouble generalizing the above for all n... Does it have something to do with saying that b_n \neq 0?
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    Re: Sequence Question

    Quote Originally Posted by FishStyx View Post
    Prove the following:

    If \{b_n\}_{n=1}^{\infty} converges to B \neq 0 and b_n \neq 0 for all n, then there is an M > 0 such that |b_n| \geq M for all n.

    Now, this sounds very similar to a Lemma that was proven in class.

    If \{b_n\}_{n=1}^{\infty} converges to B and B \neq 0, then there is a positive real number M and a positive integer N such that, if n \geq N, then |b_n| \geq M.
    Using the notation from the lemma, what can say about
    M'=\min\{M,~|b_1|,~|b_2|,\cdots,~|b_{N-1}|\}
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  3. #3
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    Re: Sequence Question

    Quote Originally Posted by Plato View Post
    Using the notation from the lemma, what can say about
    M'=\min\{M,~|b_1|,~|b_2|,\cdots,~|b_{N-1}|\}
    That M' is a lower bound of |b_n|?
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    Re: Sequence Question

    Quote Originally Posted by FishStyx View Post
    That M' is a lower bound of |b_n|?
    That is correct.

    BUT you need to point out that M'>0.
    Why is that true?
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  5. #5
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    Re: Sequence Question

    Quote Originally Posted by Plato View Post
    That is correct.

    BUT you need to point out that M'>0.
    Why is that true?
    I never really gave that much thought...

    I just supposed it was stated because M \leq |b_n| > 0 since b_n \neq 0.
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    Re: Sequence Question

    Quote Originally Posted by FishStyx View Post
    I never really gave that much thought...

    I just supposed it was stated because M \leq |b_n| > 0 since b_n \neq 0.
    Well each of these is true, M>0~\&~(\forall n)[b_n\ne 0]
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  7. #7
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    Re: Sequence Question

    Quote Originally Posted by Plato View Post
    Well each of these is true, M>0~\&~(\forall n)[b_n\ne 0]
    Hmm, yeah, I can see that.

    I'm still not quite seeing the proof yet... Am I supposed to show that there is a positive lower bound on |b_n| for all n, specifically M in the context of the problem? Or...
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