Completion of a metric space

Theorem. Given a metric space $(X,d)$, there exists a complete metric space $(\hat X,\hat d)$ and an isometric embedding $i:X\to\hat X$ with dense image. This $(\hat X,\hat d)$ is unique up to isometry.

Construction. Take equivalence classes of Cauchy sequences of $X$ (with two sequences equivalent if their pointwise distance tends to $0$).

Example. $\mathbb{Q}$ completes to $\mathbb{R}$.

Conclusion. Completion fills in "missing limits" in a canonical way.