We can assume w.l.o.g. that $Vol_{\omega}(X)=1$ and $0\leq\psi_{j}-\psi\leq 1$. Fix $\delta>0$ and $\varphi\in PSH(X,\omega)$, $0\leq\varphi\leq 1$. By Chebyshev inequality, it suffices to control $\int_{X}(\psi_{j}-\psi)\omega_{\varphi}^{n}$ uniformly in $\varphi$. It follows from Stokes theorem that

$$\int_{X}(\psi_{j}-\psi)\omega_{\varphi}^{n}=\int_{X}(\psi_{j}-\psi)\omega\wedge\omega_{\varphi}^{n-1}-\int_{X}d(\psi_{j}-\psi)\wedge d^{c}\varphi\wedge\omega_{\varphi}^{n-1}.$$

Now by Cauchy-Schwartz inequality,

$$\left|\int_{X}df_{j}\wedge d^{c}\varphi\wedge\omega_{\varphi}^{n-1}\right|\leq\left(\int_{X}df_{j}\wedge d^{c}f_{j}\wedge\omega_{\varphi}^{n-1}\right)^{1/2}\cdot\left(\int_{X}d\varphi\wedge d^{c}\varphi\wedge\omega_{\varphi}^{n-1}\right)^{1/2},$$

where we set $f_{j}:=\psi_{j}-\psi\geq 0$. Moreover

$$\int_{X}d\varphi\wedge d^{c}\varphi\wedge\omega_{\varphi}^{n-1}=\int_{X}\varphi(-dd^{c}\varphi)\wedge\omega_{\varphi}^{n-1}\leq\int_{X}\varphi\omega\wedge\omega_{\varphi}^{n-1}\leq 1,$$

since $\varphi\omega_{\varphi}^{n-1}\geq 0$, $-dd^{c}\varphi\leq\omega$ and $\varphi\leq 1$. Similarly

$$\int_{X}df_{j}\wedge d^{c}f_{j}\wedge\omega_{\varphi}^{n-1}=\int_{X}-f_{j}dd^{c}f_{j}\wedge\omega_{\varphi}^{n-1}\leq\int_{X}f_{j}\omega_{\psi}\wedge\omega_{\varphi}^{n-1}.$$

Altogether this yields

	$\displaystyle\int_{X}(\psi_{j}-\psi)\omega_{\varphi}^{n}$	$\displaystyle\leq$	$\displaystyle\int_{X}(\psi_{j}-\psi)\omega\wedge\omega_{\varphi}^{n-1}+\left(\int_{X}(\psi_{j}-\psi)\omega_{\psi}\wedge\omega_{\varphi}^{n-1}\right)^{1/2}$
		$\displaystyle\leq$	$\displaystyle\sqrt{2}\left(\int_{X}(\psi_{j}-\psi)(\omega+\omega_{\psi})\wedge\omega_{\varphi}^{n-1}\right)^{1/2},$

where the last inequality follows from the elementary inequalities $0\leq a\leq\sqrt{a}\leq 1$ and $\sqrt{a}+\sqrt{b}\leq\sqrt{2}\sqrt{a+b}$.

Going on replacing at each step a term $\omega_{\varphi}$ by $\omega+\omega_{\psi}$, we end up with

$$\int_{X}(\psi_{j}-\psi)\omega_{\varphi}^{n}\leq 2\left(\int_{X}(\psi_{j}-\psi)(\omega+\omega_{\psi})^{n}\right)^{1/2^{n}}.$$

The majorant being independent of $\varphi$ and converging to $0$ as $j\rightarrow+\infty$ (by dominated convergence theorem), this completes the proof. ∎