. Suppose that $\varphi\in\mathcal{D}(X)$ is a pointwise limit of $\theta$-psh model functions. Our goal is to show that $\varphi$ is $\theta$-psh. Upon replacing $\theta$ with $\theta+dd^{c}\varphi$ we may assume that $\varphi=0$. Note that the existence of at least one $\theta$-psh model function implies that $(\theta_{\mathcal{X}})|_{\mathcal{X}_{K}}$ is nef. As in Proposition 5.8 we can choose finitely many ample line bundles $\mathcal{A}_{i}\in\Pic(\mathcal{X})$ such that their numerical classes $\alpha_{i}\in N^{1}(\mathcal{X}/S)$ form a basis of $N^{1}(\mathcal{X}/S)$. There exists arbitrarily small positive numbers $\varepsilon=(\varepsilon_{i})$ such that $\theta_{\mathcal{X}}+\sum_{i}\varepsilon_{i}\alpha_{i}$ is a rational class, hence the class of a $\mathbf{Q}$-line bundle $\mathcal{L}_{\varepsilon}$ on $\mathcal{X}$ whose restriction to $\mathcal{X}_{K}$ is ample. Since $0$ is a pointwise limit of $\theta$-psh model functions and since $h_{\mathcal{L}_{\varepsilon}}e^{-\psi}$ is semipositive for each $\theta$-psh model function $\psi$, we may now apply Lemma 5.12 to conclude that $\mathcal{L}_{\varepsilon}$ is nef. It follows that $\theta_{\mathcal{X}}\in\Nef(\mathcal{X}/S)$ by closedness of the nef cone. ∎