By Theorem 8.7 there exists a decreasing net $(\varphi_{j})$ of $\theta$-psh model functions converging pointwise to $\varphi$. For each $\varepsilon>0$ the compact set $X$ is the increasing union of the open sets $\{\varphi_{j}<\varphi+\varepsilon\}$, hence $\varphi_{j}<\varphi+\varepsilon$ for some $j$ (Dini’s lemma). It follows that $\varphi$ lies in the closure of $\mathcal{D}(X)\cap\PSH(X,\theta)$ in $C^{0}(X)$ with respect to the topology of uniform convergence. Since the latter is defined by a norm, the result follows. ∎