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 $x_n = (1+\frac{1}{n})^n$.

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.
