Pick $\varphi\in C^{0}(X)$ vanishing on $X^{\mathrm{div}}$ and $\varepsilon>0$ rational. By Corollary 2.3 there exists a model $\mathcal{X}$ and a divisor $D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ such that $|\varphi-\varphi_{D}|\leq\varepsilon$ on $X$. The divisor $\varepsilon\mathcal{X}_{0}\pm D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ is then effective, proving $|\varphi_{D}|\leq\varepsilon$ and hence $|\varphi|\leq 2\varepsilon$ on $X$. ∎