Definition 2

A filter $\mathfrak{F}$ on a uniform space $X$ is a Cauchy filter if for each entourage $V$ of $X$ there is a subset of $X$ which is $V$-small and belongs to $\mathfrak{F}$.

Here again we may make our language more expressive by the use of the expressions "sufficiently small set" and "a set as small as we please"; thus Definition 2 can be restated by saying that a Cauchy filter is one containing arbitrarily small sets.

An infinite sequence $(u_n)$ of points of a uniform space $X$ is said to be a Cauchy sequence if the elementary filter associated with the sequence is a Cauchy filter. It comes to the same thing to say that for each entourage $V$ of $X$ there is an integer $n_0$ such that for all integers $m \geq n_0$ and $n \geq n_0$ we have $(u_m, u_n) \in V$.