Let $\varphi\in PSH(X,\omega)$. Since $X$ is projective, we can find a Hodge form $\omega^{\prime}$. Then $C^{-1}\omega^{\prime}\leq\omega\leq C\omega^{\prime}$ for some constant $C\geq 1$. Since $PSH(X,\omega)\subset PSH(X,C\omega^{\prime})$, it follows from the previous theorem that we can find $\varphi_{j}\in PSH(X,C\omega^{\prime})\cap{\mathcal{C}}^{\infty}(X)$ that decrease towards $\varphi$. Now the result follows from $PSH(X,C\omega^{\prime})\subset PSH(X,A\omega)$with $A=C^{2}$. ∎