Let $X$ be a set, let $\mathcal U$ be a cover of $X$, and let $n \in \mathbb N$. Suppose $\mathcal W = \{W_1, \ldots, W_n\}$ is a finite cover of $X$ that refines $\mathcal U$. For each $j \in \{1, \ldots, n\}$, choose $U_j \in \mathcal U$ such that $W_j \subset U_j$. Then the finite family $\{U_1, \ldots, U_n\}$ is a subcover of $\mathcal U$ covering $X$.