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Math Help - Cauchy sequences that do not converge?

  1. #1
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    Cauchy sequences that do not converge?

    Hello,

    I was under the impression that Cauchy sequences do not always converge. That is NOT true in \mathbb{R} where every Cauchy sequence converges.

    I was under the impression that there were some spaces where this is NOT true. There are some spaces where Cauchy sequences do NOT have a limit. However, I cannot find an example on the Wikipedia page:
    https://secure.wikimedia.org/wikiped...auchy_sequence

    I am asking this, because I have been reading this PDF document:
    http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf

    If you look at pages 20 and 21, you will see that at the very top of page 21, it seems to indicate that every sequence that is Cauchy has a limit.

    So what is the correct statement? Are all Cauchy sequences convergent? If yes, is the proof in the PDF a good proof? If not, do you have a counter example and why the derivation of the PDF do not work?

    I currently suspect that it has something to do with density. Like, we can find Cauchy sequences in \mathbb{Q} that have a limit in \mathbb{R}.

    Thanks in advance,
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tonybruguier View Post
    Hello,

    I was under the impression that Cauchy sequences do not always converge. That is NOT true in \mathbb{R} where every Cauchy sequence converges.

    I was under the impression that there were some spaces where this is NOT true. There are some spaces where Cauchy sequences do NOT have a limit. However, I cannot find an example on the Wikipedia page:
    https://secure.wikimedia.org/wikiped...auchy_sequence

    I am asking this, because I have been reading this PDF document:
    http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf

    If you look at pages 20 and 21, you will see that at the very top of page 21, it seems to indicate that every sequence that is Cauchy has a limit.

    So what is the correct statement? Are all Cauchy sequences convergent? If yes, is the proof in the PDF a good proof? If not, do you have a counter example and why the derivation of the PDF do not work?

    I currently suspect that it has something to do with density. Like, we can find Cauchy sequences in \mathbb{Q} that have a limit in \mathbb{R}.

    Thanks in advance,
    This is a property of whatever metric space you're working in. A metric space is called complete if it has the property that every Cauchy sequence converges. For example, \mathbb{R}^n and \mathbb{C}^n\approx\mathbb{R}^{2n} are complete for each n\in\mathbb{N}. A less trivial example is C([a,b])--the space of all continuous functions equipped with the metric induced by \|\cdot\|_{\infty}. That said, as you indicated \mathbb{Q} is not complete. There are hunderds of examples I'm sure you could think of (e.g. \left(1+\frac{1}{n}\right)^n, \displaystyle \sum_{m=1}^{n}\frac{1}{m^2}, etc.)

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