For $t>0$ large enough, the set $O_{1}=\{\varphi<-t\}$ has capacity $<\varepsilon/2$ by proposition 2.6. Working in $X\setminus O_{1}$ we can thus replace $\varphi$ by $\varphi_{t}=\max(\varphi,-t)$ which is bounded on $X$. Regularizing $\varphi$ (see Appendix), we can find a sequence $\psi_{j}$ of smooth $A\omega$-psh functions which decrease to $\varphi_{t}$ on $X$, for some $A\geq 1$. By proposition 2.7, the set $O_{j}=\{\psi_{k_{j}}>\varphi_{t}+1/j\}$ has capacity $<\varepsilon 2^{-j-1}$ if $k_{j}$ is large enough. Now $\psi_{k_{j}}$ uniformly converges to $\varphi=\varphi_{t}$ on $X\setminus O_{\varepsilon}$, $O_{\varepsilon}=\cup_{j\geq 1}O_{j}$, so $\varphi$ is continuous on $X\setminus O_{\varepsilon}$ and $Cap_{\omega}(O_{\varepsilon})\leq\varepsilon$. ∎