As above let $(\varphi_{j})_{j}$ be a decreasing net of $\omega$-psh model functions converging to $\varphi$. After adding a constant we may assume that $\varphi_{j}\leq 0$ for all $j$. For each integer $m\geq 1$, the net $(\max\{\varphi_{j},-m\})_{j}$ decreases to the bounded $\omega$-psh function $\max\{\varphi,-m\}$. We can therefore choose $j_{m}$ such that

(4.1)

$$0\leq\int\left(\max\{\varphi_{j_{m}},-m\}-\max\{\varphi,-m\}\right)\MA\left(\frac{\max\{\varphi,-m\}}{2}\right)\leq(2m)^{-2^{n+1}}.$$

We may further assume $j_{m+1}\geq j_{m}$ for all $m$. Set $\varphi_{m}:=\varphi_{j_{m}}$. We claim that the decreasing sequence $(\varphi_{m})_{m=1}^{\infty}$ converges to $\varphi$. By Theorem 2.10 it suffices to test this at any divisorial point $x\in X$. We have $0\geq\varphi(x)>-\infty$ and $\varphi_{m}(x)\geq\varphi(x)\geq-m$ for $m\geq-\varphi(x)\geq 0$. By (4.1), Lemma 4.7 and the definition of capacity we get

$$0\leq(\varphi_{m}(x)-\varphi(x))\Capa_{\omega}\{x\}\leq\frac{1}{m}$$

for $m\geq|\varphi(x)|$. Now $\Capa_{\omega}\{x\}>0$ by Lemma 4.2, thus $\varphi_{m}(x)$ converges to $\varphi(x)$, which concludes the proof. ∎