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


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

    |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 n>\max(N,N'), |a_n-a|+|b_n-b|<\epsilon/2+\epsilon/2=\epsilon

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