Set $M:=||\psi||_{L^{\infty}(X)}$. Observe that when $t\geq 0$, $(\varphi-\psi<-s-t\psi)\subset(\varphi-\psi<-s+tM)$, hence it follows from lemma 2.2 that for all $s>0$, $0\leq t\leq 1$,

(3)

$$t^{n}Cap_{\omega}(\varphi-\psi<-s-t)\leq(1+M)^{n}\int_{(\varphi-\psi<-s)}\omega_{\varphi}^{n},$$

using the obvious substitutions $s\mapsto s-Mt$, $t\mapsto(1+M)t$. Since $\mu=\omega_{\varphi}^{n}$ satisfies ${\mathcal{H}}(\alpha,A,\omega)$, we infer

$$t^{n}Cap_{\omega}(\varphi-\psi<-s-t)\leq A(1+M)^{n}Cap_{\omega}(\varphi-\psi<-s)^{1+\alpha}.$$

Consider

$$f(s):=\left[Cap_{\omega}(\varphi-\psi<-s-\varepsilon)\right]^{1/n},\,s>0.$$

Then $f$ satisfies the condition $H(\alpha,B)$ of lemma 2.3 with $B=(1+M)A^{1/n}$. Assume $f(0)=\left[Cap_{\omega}(\varphi-\psi<-\varepsilon)\right]^{1/n}<\frac{1}{(2B)^{1/\alpha}}$. It follows in this case from Remarks 2.4 that $f(s)=0$ for $s\geq S_{\infty}$, where

$$S_{\infty}\leq\frac{2B}{1-2^{-\alpha}}\left[Cap_{\omega}(\varphi-\psi<-\varepsilon)\right]^{\alpha/n}$$

Therefore the sets $\{\varphi-\psi<-s-\varepsilon\}$ are empty for $s>S_{\infty}$, hence

$$\sup_{X}(\psi-\varphi)\leq\varepsilon+S_{\infty}\leq\varepsilon+\frac{2(1+M)A^{1/n}}{1-2^{-\alpha}}\left[Cap_{\omega}(\varphi-\psi<-\varepsilon)\right]^{\alpha/n}.$$

Thus we can take here $C\geq 2B/(1-2^{-\alpha})$.

If $f(0)=[Cap_{\omega}(\varphi-\psi<-\varepsilon)]^{1/n}\geq(2B)^{-1/\alpha}$, then

$$\sup_{X}(\psi-\varphi)\leq-\inf_{X}\varphi\leq C_{2}(\alpha,A,\omega),$$

by (2), hence it suffices to take $C\geq 2BC_{2}$ to conclude. ∎