For completeness, does the limit of the Cauchy sequence need to be in the same space as the sequence?

Question

I know $\mathbb{R}$ is complete since every Cauchy sequence of numbers has a limit. But does this limit need to be in the same metric space as the sequence. For example is $\mathbb{Q}$ complete? Every Cauchy sequence has a limit, but not necessarily in $\mathbb{Q}$, just take $x_n = (1+\frac{1}{n})^n$.

Answer 1

No, $\mathbb{Q}$ is not complete because it does not contain all its limit points.

It is easy to transform any metric space into a complete metric space by adding in the limit points. More precisely, one can consider the set of Cauchy sequences on a metric space under the equivalence relation that two Cauchy sequences are the same if their difference converges to $0$. This is one way of completing a space. For instance, the real numbers can be constructed from $\mathbb{Q}$ in this way.

Answer 2

Yes. Generally, something is complete if it doesn't need anything added to it. In this particular sense of complete, $\Bbb R$ is complete because limit points for Cauchy sequences in $\Bbb R$are already in $\Bbb R$ and so do not need to be added, while $\Bbb Q$ is not complete because limit points for Cauchy sequences in $\Bbb Q$ do not in general belong to $\Bbb Q$.