Proposition 2

On a uniform space $X$ every convergent filter is a Cauchy filter.

If $x$ is any point of $X$ and $V$ is any symmetric entourage of $X$, then the neighbourhood $V(x)$ of $x$ is $\overline{V}$-small. If $\mathfrak{F}$ is a filter which converges to $x$, there is a set of $\mathfrak{F}$ contained in $V(x)$, and therefore $V$-small.

Clearly every filter which is finer than a Cauchy filter is a Cauchy filter.