Recall that the space of model functions ${\mathscr{D}}(X)$ is dense in $C^{0}(X^{\mathrm{an}})$ for the topology of uniform convergence [Gub98, Thm. 7.12]. Hence we may assume that $u\in{\mathscr{D}}(X)$ by Proposition 2.9(v). Observe that by Proposition 2.9(vii), we may replace $(\theta,u)$ by a suitable multiple. Hence we may assume without loss of generality that the model function $u$ is defined by a vertical divisor on a ${K^{\circ}}$-model $\mathscr{X}^{\prime}$. It is clear that we can choose $\mathscr{X}^{\prime}$ dominating the geometric model $\mathscr{X}=\mathscr{X}_{R}\otimes_{R}{K^{\circ}}$ of $\theta$ from Definition 8.1. It follows from Proposition 5.2(a) that we may assume $\mathscr{X}=\mathscr{X}^{\prime}$. By Proposition 2.9(iv) we get

(8.1)

$${P}_{\theta}(u)={P}_{\theta+dd^{c}u}(0)+u.$$

By construction, the class $\theta+dd^{c}u$ is induced by a line bundle on $\mathscr{X}_{R}$ and hence Corollary 7.4 yields that ${P}_{\theta+dd^{c}u}(0)$ is a uniform limit of $(\theta+dd^{c}u)$-psh model functions $\varphi_{i}$. Then $P_{\theta}(u)$ is the uniform limit of the sequence of $\theta$-psh functions $\varphi_{i}+u$ by (8.1). ∎