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So I'm studying completeness of metric spaces, and it seems that most incomplete spaces are subsets of another space which contains the limits of their Cauchy sequences. I was wondering if it were possible for an incomplete metric space to have a Cauchy sequence that has no limit in any space whatsoever, including any space that could contain that incomplete space. Thanks.

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    $\begingroup$ Every metric space has a completion. $\endgroup$ – Lord Shark the Unknown Sep 7 at 5:01
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    $\begingroup$ Haha. I'm literally pages away from the section on completion, I guess I should have waited. $\endgroup$ – Aphyd Sep 7 at 5:02
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It so happens that every metric space $(X,d_X)$ has a completion, $(\hat{X},d_{\hat{X}})$ which comes equipped with an embedding $i:X\hookrightarrow{} \hat{X}$ so that $d_{\hat{X}}(i(x),i(x'))=d_X(x,x').$ The construction essentially "formally" adds the limits of Cauchy sequences by defining the elements of $\hat{X}$ to be Cauchy sequences $(x_n)$ of elements of $X$ modulo the equivalence relation stating that $(x_n)\sim (y_n)$ if and only if $\lim_{n\to\infty} d_X(x_n,y_n)=0$. The elements of $X$ can be identified with the equivalence classes of the constant sequences $(x_n=x)$.

Actually, applying this construction to Cauchy sequences with respect to the usual norm on $\mathbb{Q}$ gives us the field $\mathbb{R}$. Applying this construction to $\mathbb{Q}$ with the $p-$adic norm $\lvert \:\cdot\:\rvert_p$ produces the $p-$adic numbers $\mathbb{Q}_p$.

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Good thinking. But you should go one step further: given an incomplete space, if you don't know whether it is contained in a larger metric space, why not make one yourself?

The process is called completion. To put it simply, you see every Cauchy sequence as a single object, and define the equivalence classes of Cauchy sequences. All these equivalence classes constitue a new space that is complete. The original space can be seen as a subspace of (embedded into) the new space.

Since you can always do completion. The answer to you question is: there's always a larger space that's complete.

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