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Math Help - Convergence

  1. #1
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    Convergence

    [a_n] and [b_n] be two sequences

    Let Limit n->oo ( a_n-b_n) = 0 (i.e Limit n tends to infinity, an=bn)

    Is Limit n->oo ( a_n^2 -b_n^2) =0?


    (I'm trying to prove that there might exist nested rational interval with no rational point - I need the above for that. Essentially I have setup a_n^2<2 and b_n^2>2, thus will show that if x belongs to this interval then x^2=2 => x is not rational)

    Thanks
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  2. #2
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    Quote Originally Posted by aman_cc View Post
    [a_n] and [b_n] be two sequences
    Let Limit n->oo ( a_n-b_n) = 0 (i.e Limit n tends to infinity, an=bn)
    Is Limit n->oo ( a_n^2 -b_n^2) =0?
    If you know that the two sequences are both bounded, then yes the result follows.
    From what you said you are doing, it seems the sequences would be bounded.

    Just notice that |a^2-b^2|=|a-b||a+b|.
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  3. #3
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    Quote Originally Posted by Plato View Post
    If you know that the two sequences are both bounded, then yes the result follows.
    From what you said you are doing, it seems the sequences would be bounded.

    Just notice that |a^2-b^2|=|a-b||a+b|.
    Thanks Plato. Indeed they are bounded.
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