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Math Help - Real analysis - limit proofs

  1. #1
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    Real analysis - limit proofs

    Let a(n) and b(n) be convergent sequences of real numbers such that the limit of a(n) as n approaches infinity is A and the limit of b(n) as n approaches infinity is B for some real numbers A and B. Show (using the epsilon-N definition of a limit) that

    (a) the limit of a(n) + b(n) as n approaches infinity = A + B

    (b) the limit of a(n)b(n) as n approaches infinity = AB
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  2. #2
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    The following are hints.

    Quote Originally Posted by alexmin View Post
    (a) the limit of a(n) + b(n) as n approaches infinity = A + B
    |a_n + b_n - (a+b)| = |(a_n-a)+(b_n-b)| \leq |a_n-a|+|b_n-b|

    (b) the limit of a(n)b(n) as n approaches infinity = AB
    Since a_n \to a the sequence is bounded, so |a_n|\leq M for M>0.
    This means,
    |a_nb_n - ab| = |a_nb_n - a_nb + a_nb - ab| \leq |a_n||b_n - b|+ |b||a_n - a| \leq M|b_n-b|+b|a_n-a|.
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