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Not all sequences that are Cauchy are convergent. Here is what I think the example should be. Somehow the metric space is open but does not contain its limit points. Is this the right direction of thought?

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What is the definition of a "convergent space" and a "Cauchy space"? Do you mean "sequence" instead of "space"? In that case, try constructing a space in which nothing is Cauchy? –  akkkk Oct 28 '12 at 15:57
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The terms Cauchy and convergent refer to sequences, not spaces. Are you asking for a metric space in which there are Cauchy sequences that do not converge? One familiar example is $\Bbb Q$. –  Brian M. Scott Oct 28 '12 at 15:57
    
@akkkk: A constant sequence is Cauchy in any metric space. –  Brian M. Scott Oct 28 '12 at 15:58
    
yes yes, sorry! i am asking for a space that has Cauchy sequences that does not converge, so to show that there should be a subset of that space which is open and therefore doesnt contain limit points, hence does not converge. –  d13 Oct 28 '12 at 16:00
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@d13: You should be able to edit your question and make this correction. –  Nate Eldredge Oct 28 '12 at 16:04
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up vote 7 down vote accepted

Just take any sequence of rational numbers that converges to an irrational number. Then the sequence is Cauchy in $ \mathbb{Q} $, but does not converge in $ \mathbb{Q} $.

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There can be sequences that are Cauchy but do not converge; for example, the sequence $(1,\frac{1}{2},\frac{1}{3},\ldots)$ does not converge in the metric space $(0,2)$. A metric space in which every Cauchy sequence converges is said to be complete.

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In $\mathbb{K}[X]$.

$\forall P \in \mathbb{K}[X], P = \sum\limits_{i \in\mathbb{N}} p_i X^i, \|P\|=\max\limits_{i \in\mathbb{N}}( | p_i | )$

Let $P_n = \sum\limits_{i = 1}^n \frac{X^i}{i!}$

$P_n$ is Cauchy but doesn't converge in $\mathbb{K}[X]$.

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