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Math Help - cauchy sequence

  1. #1
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    cauchy sequence

    Prove that if  f:A \to R  is uniformly continuous on A and  x_n  is a cauchy sequence in A then  f(x_n)  is a cauchy sequence in R .
    R=real numbers
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by flower3 View Post
    Prove that if  f:A \to R  is uniformly continuous on A and  x_n  is a cauchy sequence in A then  f(x_n)  is a cauchy sequence in R .
    R=real numbers
    Fix \epsilon>0. Then by uniform continuity, \exists\delta>0 s.t. |x_n-x_m|<\delta \implies |f(x_n)-f(x_m)|<\epsilon.

    Since \{x_n\} is Cauchy, \forall\delta>0, \exists N s.t. n,m>N \implies |x_n-x_m|<\delta

    So...
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