Corollary 2

Every cluster point $x$ of a Cauchy filter $\mathfrak{F}$ is a limit point of $\mathfrak{F}$.

There is a filter $\mathfrak{G}$ which is finer than both $\mathfrak{F}$ and $\mathfrak{B}(x)$ (Chapter I, § 7, no. 2, Proposition 4); since $\mathfrak{F}$ is a Cauchy filter, so is $\mathfrak{G}$. If $\mathfrak{F}_0$ is the unique minimal Cauchy filter coarser than $\mathfrak{F}$, then both $\mathfrak{F}_0$ and $\mathfrak{B}(x)$ are minimal Cauchy filters coarser than $\mathfrak{G}$. Hence $\mathfrak{F}_0 = \mathfrak{B}(x)$, which shows that $\mathfrak{F}$ converges to $x$.