Example 2

Let $X$ be an infinite set, and consider the uniformity of finite partitions on $X$ ($\S$ 2, no. 2, Remark 1). Every ultrafilter $\mathfrak{F}$ on $X$ is a

Cauchy filter with respect to this uniformity. For if $(A_i)$ is a finite partition of $X$ and

the corresponding entourage, then at least one of the $A_i$ belongs to $\mathfrak{F}$ (Chapter I, § 6, no. 4, Corollary to Proposition 5), and $A_i$ is V-small. On the other hand, $X$ is an infinite discrete space, hence is not compact, and consequently there are ultrafilters on $X$ which do not converge.