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Math Help - Cauchy sequence/Continuity Question

  1. #1
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    Cauchy sequence/Continuity Question

    Let f: (a,b] \rightarrow \mathbb{R} be a real valued continuous function defined on the half open interval (a,b]. Assume that f has the property than whenever (x_n) is a cauchy sequence contained in the open interval (a,b], then (f(x_n)) is a cauchy sequence.
    Prove that the one sided limit:
    \lim_{x \to a+} f(x)exists and deduce that f is uniformly continuous.

    So far all I have is the definition of a cauchy sequence and the definition of a right-hand sided limit, but can't see how to start this question. Help?
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    MHF Contributor Drexel28's Avatar
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    Re: Cauchy sequence/Continuity Question

    Quote Originally Posted by worc3247 View Post
    Let f: (a,b] \rightarrow \mathbb{R} be a real valued continuous function defined on the half open interval (a,b]. Assume that f has the property than whenever (x_n) is a cauchy sequence contained in the open interval (a,b], then (f(x_n)) is a cauchy sequence.
    Prove that the one sided limit:
    \lim_{x \to a+} f(x)exists and deduce that f is uniformly continuous.

    So far all I have is the definition of a cauchy sequence and the definition of a right-hand sided limit, but can't see how to start this question. Help?
    The idea is to prove that you can continuously extend f: (a,b] to \widetilde{f}:[a,b] by defining \displaystyle f(a)=\lim_{x\to a^+}f(x). You can then conclude from the Heine-Cantor theorem that \widetilde{f} is unif. cont. and since the restriction of unif. cont. maps are unif. cont. you'd have that f is unif. cont.
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