# Is it possible for a Cauchy sequence to have no limit point at all?

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.

• Every metric space has a completion. – Lord Shark the Unknown Sep 7 at 5:01
• Haha. I'm literally pages away from the section on completion, I guess I should have waited. – Aphyd Sep 7 at 5:02

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 $$\mathbb{Q}_p$$.