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Math Help - Cauchy Sequences

  1. #1
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    Cauchy Sequences

    Problems:
    For each positive integer k define

    a. f_{k}(x)=x^k
    b. f_{k})(x)=e^{x/k}
    c. f_{k}(x)=cos(x/k)

    for 0 \leq x \leq 1. Is the sequence { f_{k}:[0,1] \rightarrow R} a Cauchy Sequence in the metric space C([0,1]),R

    =====================================
    By definition, a sequence  {f_{k}} is Cauchy sequence in C([0,1]),R if and only if there is an  \epsilon > 0

    such that

    |f_{k}(x)-f_{l}(x) | < \epsilon for all x in [a, b]

    For a I said that f_{k}(x)=x^k is only Cauchy for when  k < 1 since the distance between the f_{k}'s get smaller and smaller together. Thus this sequence is a Cauchy sequence.

    For b, f_{k})(x)=e^{x/k}, I said that this is not Cauchy as it is distance is getting larger and larger since the sequence of function grows exponentially. So as k and l goes to infinity, |f_{k}(x)-f_{l}(x) |  does not go to zero.

    For c, f_{k}(x)=cos(x/k), I said that this is Cauchy with a similar argument with part (a).

    I have no idea how to put this in a formal proof though, but the problem seems like it only wants a true or false type of answer, I think.

    Thank you for your time.
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  2. #2
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    Quote Originally Posted by Paperwings View Post
    a. f_{k}(x)=x^k
    If this was a uniform Cauchy sequence on [0,1] then it would be a uniformly convergent sequence on [0,1]. If it was a uniformly convergent sequence on [0,1] then the limit function would be continous. But it is not.
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