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  1. #1
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    not cauchy

    Give an example of a continuous function  f: A \to  \mathbb{R} on a subset A of    \mathbb{R} and a Cauchy sequence x_n in A in which  f(x_n) is not a cauchy sequence in     \mathbb{R}
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    MHF Contributor Bruno J.'s Avatar
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    Take 0,1] \longrightarrow \mathbb{R}" alt="f0,1] \longrightarrow \mathbb{R}" /> given by x \mapsto x^{-1} and take x_n=n^{-1}. Then f(x_n)=n is certainly not Cauchy.
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  3. #3
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    Crucial point- this sequence converges to a point that is NOT in A. If the x_n converged to a point, a, in A, since f is continuous on A, f(x_n) would have to converge to f(a), and so be Cauchy.
    Last edited by HallsofIvy; December 30th 2009 at 08:34 AM.
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