Example

In any uniform space $X$, the set of points of a Cauchy sequence $(x_n)$ is precompact. Since the images of the $x_n$ in $\hat{X}$ again form a Cauchy sequence, we can assume that $X$ is Hausdorff; the closure in $\hat{X}$ of the set of points $x_n$ then consists of the points $x_n$ and $\lim_{n \to \infty} x_n$ and is therefore compact (Chapter I, § 9, no. 3, Example 2).