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Math Help - a simple limit-series problem

  1. #1
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    a simple limit-sequence problem

    lim(a_n)=a, lim(b_n)=b

    Prove:

    if a<b, then there is an index  n_0 so that for every n>=n_0 : a_n < b_n

    This one should be really easy, I just have no idea how to look at it... Any tips for calculus will be absolutely great!
    Last edited by adam63; November 28th 2009 at 02:36 AM.
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  2. #2
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    Quote Originally Posted by adam63 View Post
    lim(a_n)=a, lim(b_n)=b

    Prove:

    if a<b, then there is an index  n_0 so that for every n>=n_0 : a_n < b_n

    This one should be really easy, I just have no idea how to look at it... Any tips for calculus will be absolutely great!

    Take \epsilon:=\frac{b -a}{2}\,\Longrightarrow\,\,\exists N_1\,,\,N_2\in\mathbb{N}\,\,s.t.\,\,|a_n-a|<\epsilon\,\,\forall n>N_1\,,\,\,|b_n-b|<e\,\,\forall n>N_1 \Longleftrightarrow -\epsilon<a_n-a<\epsilon\,,\,-\epsilon<b_n-b<\epsilon

    \Longrightarrow\,\forall n>max(N_1\,,\,N_2)\,,\,b_n-a_n=b_n-b+b-a-(a_n-a)>-2\epsilon+b-a>0 ...

    Tonio
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  3. #3
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    Thank you very much !!
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