Proposition 5.4.4 (Cauchy Criterion). Let $(X, \fU)$ be a uniform space and $\fF \subset 2^{X}$ be a convergent filter, then $\fF$ is Cauchy.
Proof. Let $x \in X$ such that $\cn(x) \subset \fF$ and $V \in \fU$. By Lemma 5.1.9, assume without loss of generality that $V \in \fU$. Since $\cn(x) \subset \fF$, then $V(x) \in \fU$ and $V(x) \times V(x) \subset V$.$\square$