Since we may check semipositivity after a base extension (see Lemma 3.3), we may replace $K$ by a finite field extension of $K$. Then, using Lemma 3.6, we may assume that $X$ is a variety.

Let $\varphi$ be a model function on $X$ which is the pointwise limit of $\theta$-psh functions. Replacing $\theta$ by $\theta+dd^{c}\varphi$, we may assume that $\varphi=0$. Then the existence of a $\theta$-psh function yields that $\theta$ is semipositive and hence $\{\theta\}$ is nef (see 4.8). Let ${\mathscr{X}}$ be a ${K^{\circ}}$-model of $X$ such that $\theta$ is determined on ${\mathscr{X}}$. Then the restriction of $\theta_{\mathscr{X}}$ to $X$ is nef.

We may replace ${\mathscr{X}}$ by a generically finite covering ${\mathscr{X}}^{\prime}$ for any ${K^{\circ}}$-model ${\mathscr{X}}^{\prime}$ with generic fibre $X^{\prime}$. This does not change convergence of metrics and semipositivity. It is here, where we use that pointwise convergence holds on ${X^{\rm an}}$. By [dJ96, Theorem 4.5], up to replacing $K$ by a finite field extension, we may assume that ${\mathscr{X}}$ is SNC (see 5.1). The proof of Proposition 4.13 shows that $N^{1}({\mathscr{X}}/S)$ is a finite dimensional ${\mathbb{R}}$-vector space as we can see it as a subspace of $N^{1}({\mathscr{X}}_{s})$. We have also seen that the ample cone in $N^{1}({\mathscr{X}}/S)$ is the intersection of $N^{1}({\mathscr{X}}/S)$ with the ample cone in $N^{1}({\mathscr{X}}_{s})$ and hence it is open in $N^{1}({\mathscr{X}}/S)$. We conclude that there are ${\mathscr{H}}_{1},\dots,{\mathscr{H}}_{n}$ ample line bundles on ${\mathscr{X}}$ such that their numerical classes $\alpha_{j}$ form a basis of $N^{1}({\mathscr{X}}/S)$. Then there are $\lambda_{j}\in{\mathbb{R}}$ such that

$$c_{1}({\mathscr{L}}):=\sum_{j}\lambda_{j}c_{1}({\mathscr{H}}_{j})\in{\rm Pic}({\mathscr{X}})_{\mathbb{R}}$$

represents $\theta$. Let ${\varepsilon}_{j}$ be small positive numbers such that the numbers $\lambda_{j}+{\varepsilon}_{j}$ are rational. We consider the ${\mathbb{Q}}$-line bundle

$${\mathscr{L}}_{\varepsilon}:=\bigotimes_{j}{\mathscr{H}}_{j}^{\otimes(\lambda_{j}+{\varepsilon}_{j})}$$

on ${\mathscr{X}}$ and let $L_{\varepsilon}:={\mathscr{L}}_{\varepsilon}$. Since $\{\theta\}$ is nef and ${\varepsilon}_{j}>0$, it follows that $L_{\varepsilon}$ is ample. For any model function $\psi$ on $X$, we have

$$c_{1}(L_{\varepsilon},e^{-\psi}{\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}})=dd^{c}\psi+\theta+\sum_{j}{\varepsilon}_{j}\alpha_{j}.$$

We conclude that a $\theta$-psh model function $\psi$ yields a semipositive model metric $e^{-\psi}{\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}}$. Since $0$ is the pointwise limit of $\theta$-psh model functions $\psi$, we deduce that ${\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}}$ is the pointwise limit of semipositive model metrics on $L_{\varepsilon}$. It follows from Proposition 5.2 that ${\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}}$ is semipositive. This means that ${\mathscr{L}}_{\varepsilon}$ is nef.

By definition of nef and using $N^{1}({\mathscr{X}}/S)\subset N^{1}({\mathscr{X}}_{s})$, we see that the cone in $N^{1}({\mathscr{X}}/S)$ of nef classes is the intersection of $N^{1}({\mathscr{X}}/S)$ with the nef cone in $N^{1}({\mathscr{X}}_{s})$. In particular, the cone of nef classes is closed in $N^{1}({\mathscr{X}}/S)$. Using ${\varepsilon}=({\varepsilon}_{1},\dots,{\varepsilon}_{n})\to 0$, we deduce that ${\mathscr{L}}$ is nef. Since ${\mathscr{L}}$ represents $\theta$, we conclude that $\varphi=0$ is $\theta$-psh. ∎