Consider the Ricci-flat metrics $\tilde{\omega}_{t}$ on $X$, which can be written as $\tilde{\omega}_{t}=\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$. As $t\to 0$ we have that $\varphi_{t}\to\psi$ in the $C^{1,\beta}_{loc}$ topology on $X\backslash S$, for any $0<\beta<1$, and so $\tilde{\omega}_{t}$ converges in this topology to $\omega$, which satisfies (4.4). Moreover $\tilde{\omega}_{t}$ also converge to $\omega$ weakly as currents on $X$.