Let $(\varphi_{j})_{j}$ be a decreasing net of $\omega$-psh model functions converging to $\varphi$. We may assume that $-M\leq\varphi_{j}\leq 0$ for all $j$, where $M\geq 1$. For any $\omega$-psh function $\psi$ with $-1\leq\psi\leq 0$ it follows from Lemma 4.7 that

$$0\leq\int(\varphi_{j}-\varphi)\MA(\psi)\leq 4M\left(\int(\varphi_{j}-\varphi)\MA(\frac{\varphi}{2})\right)^{\frac{1}{2^{n}}}$$

and the right hand side tends to zero as $j\to\infty$ by Theorem 3.1. It therefore follows from the definition of the capacity and from Chebyshev’s inequality that for each integer $m\geq 1$ there exists $j_{m}$ such that the open set $G_{m}:=\{\varphi_{j_{m}}-\varphi>\frac{1}{m}\}$ has capacity $2^{-m}\varepsilon$. We can then set $G:=\bigcup_{m}G_{m}$ and $\varphi_{m}:=\varphi_{j_{m}}$. ∎