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Thread: Sequence Question

  1. #1
    Newbie FishStyx's Avatar
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    Sequence Question

    Prove the following:

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

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

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

    I'm just having trouble generalizing the above for all n... Does it have something to do with saying that $\displaystyle b_n \neq 0$?
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  2. #2
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    Re: Sequence Question

    Quote Originally Posted by FishStyx View Post
    Prove the following:

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

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

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

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

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

    I just supposed it was stated because $\displaystyle M \leq |b_n| > 0$ since $\displaystyle b_n \neq 0$.
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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 $\displaystyle M \leq |b_n| > 0$ since $\displaystyle b_n \neq 0$.
    Well each of these is true, $\displaystyle M>0~\&~(\forall n)[b_n\ne 0]$
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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, $\displaystyle M>0~\&~(\forall n)[b_n\ne 0]$
    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 $\displaystyle |b_n|$ for all $\displaystyle n$, specifically $\displaystyle M$ in the context of the problem? Or...
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