Let $\mathcal{L}$ be a language and let $T$ and $T^{\prime}$ be $\mathcal{L}$-theories. Suppose that for every model $\mathcal{M}$ of $T$ there exists $\sigma \in T^{\prime}$ such that $\mathcal{M} \models \sigma$. How to show that there exists a finite subset $\{\sigma_1,\ldots,\sigma_n\}$ of $T^{\prime}$ such that $T \models \sigma_1 \vee \cdots \vee \sigma_n$ ?
I was thinking to look at $T \cup \{ \neg \sigma \mid \sigma \in T^{\prime}\}$, because we know that $T \models \sigma$, so $T \cup \{ \neg \sigma \}$ is inconsistent, and then to apply Compactness theorem II. Something like that.
By assumption $T\cup\{\neg\sigma:\sigma\in T'\}$ is inconsistent. By compcatness $T\cup\{\neg\sigma_1\dots,\neg\sigma_n\}$ is inconsistent for some $\sigma_1,\dots,\sigma_n\in T'$. Hence $T\models\sigma_1 \vee \cdots \vee \sigma_n$