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Thread: whats the rule about..

  1. #1
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    whats the rule about..

    whats the rule about the sum of to convergent sequences??
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  2. #2
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    Quote Originally Posted by transgalactic View Post
    whats the rule about the sum of to convergent sequences??
    Please, can't you be more careful when you write something ???

    it's not "to", it's "two"... without context, it would be hard to guess



    Let $\displaystyle a_n \to a$ : this means that $\displaystyle \forall \epsilon>0, \exists N, \forall n>N, |a_n-a|<\epsilon/2$
    similarly, $\displaystyle b_n \to b$ : $\displaystyle \forall \epsilon>0, \exists N', \forall n>N', |a_n-a|<\epsilon/2$


    now what about $\displaystyle a_n$ and $\displaystyle b_n$ ? do they converge to $\displaystyle a+b$ ?

    $\displaystyle |a_n+b_n-(a+b)|=|(a_n-a)+(b_n-b)| \leq |a_n-a|+|b_n-b|$, by the triangle inequality.
    And for $\displaystyle n>\max(N,N')$, $\displaystyle |a_n-a|+|b_n-b|<\epsilon/2+\epsilon/2=\epsilon$

    this is the definition for $\displaystyle a_n+b_n \to a+b$
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