For completeness, does the limit of the Cauchy sequence need to be in the same space as the sequence? 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 $x_n = (1+\frac{1}{n})^n$.

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 $\Bbb Q$.
