Lemma 5.4.5. Let $(X, \fU)$ be a uniform space and $\net{x}\subset X$ be a Cauchy net. If there exists a subnet $\angles{x_\beta}_{B} \subset \net{x}$ and $x \in X$ such that $x_{\beta} \to x$, then $x_{\alpha} \to x$.
Proof. Let $U \in \fU$ and $V \in \fU$ such that $V \circ V \subset U$, then there exists $\alpha_{0} \in A$ such that $(x_{\alpha}, x_{\alpha'}) \in V$ for all $\alpha, \alpha' \ge \alpha_{0}$, and $\beta \in B$ with $\beta \ge \alpha_{0}$ such that $(x_{\beta}, x) \in V$. Thus, $(x_{\alpha}, x) \in V \circ V \subset U$ for all $\alpha \ge \alpha_{0}$, so $x_{\alpha} \to x$.$\square$