Let $\varphi\in PSH(X,\omega)$. We can assume w.l.o.g. that $\varphi\leq 0$ on $X$. Let $\psi=\{\psi_{\alpha}:=\varphi+h_{\alpha}\in PSH({\mathcal{U}}_{\alpha})\}$ denote the associated (singular) positive metric of $L$ on $X$, where $\{{\mathcal{U}}_{\alpha}\}$ denotes an open cover of $X$ trivializing $L$ (see section 4).

Step 1. We consider the following Bergman spaces

$${\mathcal{H}}_{j,j_{0}}:=\left\{s\in\Gamma(X,L^{j})\,/\,\int_{X}|s|^{2}e^{-2h_{j,j_{0}}}dV_{\omega}<+\infty\right\},$$

where $h_{j,j_{0}}=(j-j_{0})\psi+j_{0}h$, $j_{0}$ a fixed large integer (to be specified later). Let $\sigma_{1}^{(j,j_{0})},\ldots,\sigma_{s_{j}}^{(j,j_{0})}$ be an orthonormal basis of ${\mathcal{H}}_{j,j_{0}}$ and set

$$\psi_{j,j_{0}}:=\frac{1}{2j}\log\left[\sum_{l=1}^{s_{j}}|\sigma_{l}^{(j,j_{0})}|^{2}\right]=\frac{1}{2j}\sup_{s\in B_{j,j_{0}}}\log|s|^{2},$$

where $B_{j,j_{0}}$ denotes the unit ball of radius 1 centered at 0 in ${\mathcal{H}}_{j,j_{0}}$. Clearly $\psi_{j,j_{0}}$ defines a positive (singular) metric of $L$ on $X$, equivalently $\varphi_{j,j_{0}}:=\psi_{j,j_{0}}-h\in PSH(X,\omega)$. If $x\in{\mathcal{U}}_{\alpha}$ and $s=\{s_{\alpha}\}\in{\mathcal{H}}_{j,j_{0}}$, then $|s_{\alpha}|^{2}$ is subharmonic in ${\mathcal{U}}_{\alpha}$ hence

$$|s_{\alpha}(x)|^{2}\leq\frac{C_{1}}{r^{2n}}\int_{B(x,r)}|s_{\alpha}(x)|^{2}\leq\frac{C_{2}}{r^{2n}}e^{2\sup_{B(x,r)}h_{j,j_{0}}}\int_{X}|s|^{2}e^{-2h_{j,j_{0}}}dV_{\omega},$$

where $r>0$ is so small that $B(x,r)\subset{\mathcal{U}}_{\alpha}$. We infer

(3)

$$\varphi_{j,j_{0}}(x)\leq(1-j_{0}/j)\sup_{B(x,r)}\varphi+\frac{C_{3}-n\log r}{j}.$$

There is also a reverse inequality which uses a deep extension result of Ohsawa-Takegoshi-Manivel (see [14]): there exists $j_{0}\in\mathbb{N}$ and $C_{4}>0$ large enough so that $\forall x\in X,\forall j\in\mathbb{N}$, there exists $s\in\Gamma(X,L^{j})$ with

$$\int_{X}|s|^{2}e^{-2h_{j,j_{0}}}dV_{\omega}\leq C_{4}|s(x)|^{2}e^{-2h_{j,j_{0}}(x)}.$$

Choose $s$ so that the right hand side is equal to $1$, hence $s\in B_{j,j_{0}}$. Then

$$\psi_{j,j_{0}}(x)\geq\frac{1}{2j}\log|s(x)|^{2}=\left(1-\frac{j_{0}}{j}\right)\psi(x)+\frac{j}{j_{0}}h(x)-\frac{\log C_{4}}{2j}.$$

We infer

(4)

$$\varphi_{j,j_{0}}(x)\geq\left(1-\frac{j_{0}}{j}\right)\varphi(x)-\frac{\log C_{4}}{2j}\geq\varphi(x)-\frac{\log C_{4}}{2j}$$

since $\varphi\leq 0$ on $X$. It follows from (3) and (4) that $\varphi_{j}\rightarrow\varphi$ in $L^{1}(X)$.

Step 2. We now show, following [16] that $(\varphi_{j,j_{0}})_{j}$ is almost subadditive. Let $s\in\Gamma(X,L^{j_{1}+j_{2}})$ with

$$\int_{X}|s|^{2}e^{-2h_{j_{1}+j_{2},j_{0}}}dV_{\omega}\leq 1.$$

We may view $s$ as the restriction to the diagonal $\Delta$ of $X\times X$ of a section $S\in\Gamma(X\times X,L_{1}^{j_{1}}\otimes L_{2}^{j_{2}})$, where $L_{i}=\pi_{i}^{*}L$ and $\pi:X\times X\rightarrow X$ denotes the projection onto the $i^{th}$ factor, $i=1,2$. Consider the Bergman spaces

	$\displaystyle{\mathcal{H}}_{j_{1},j_{2},j_{0}}:=\left\{S\in\Gamma(X\times X,L_{1}^{j_{1}}\otimes L_{2}^{j_{2}})\,/\,\right.$
			$\displaystyle\left.\int_{X\times X}|S|^{2}e^{-2h_{j_{1},j_{0}/2}(x)-2h_{j_{2},j_{0}/2}(y)}dV_{\omega_{1}}(x)dV_{\omega_{2}}(y)<+\infty\right\},$

where $\omega_{i}=\pi^{*}\omega$. It follows from the Ohsawa-Takegoshi-Manivel $L^{2}$-extension theorem [15] that there exists $S\in\Gamma(X\times X,L_{1}^{j_{1}}\otimes L_{2}^{j_{2}})$ such that $S_{|\Delta}=s$ and

$$\int_{X\times X}|S|^{2}e^{-2h_{j_{1},j_{0}/2}-2h_{j_{2},j_{0}/2}}dV_{\omega_{1}}dV_{\omega_{2}}\leq C_{5}\int_{X}|s|^{2}e^{-2h_{j_{1}+j_{2},j_{0}}}dV_{\omega}\leq C_{5},$$

where $C_{5}$ only depends on the dimension $n=\dim_{\mathbb{C}}X$. Observe that $\{\sigma_{l_{1}}^{(j_{1},j_{0}/2)}(x)\cdot\sigma_{l_{1}}^{(j_{1},j_{0}/2)}(y)\}_{l_{1},l_{2}}$ forms an orthonormal basis of ${\mathcal{H}}_{j_{1},j_{2},j_{0}}$, thus

$$S(x,y)=\sum_{l_{1},l_{2}}c_{l_{1},l_{2}}\sigma_{l_{1}}^{(j_{1},j_{0}/2)}(x)\sigma_{l_{2}}^{(j_{2},j_{0}/2)}(y)$$

with $\sum|c_{l_{1},l_{2}}|^{2}\leq C_{5}$. It follows therefore from Cauchy-Schwarz inequality that

$$|s(x)|^{2}=|S(x,x)|^{2}\leq C_{5}\sum_{l_{1}}|\sigma_{l_{1}}^{(j_{1},j_{0}/2)}(x)|^{2}\sum_{l_{2}}|\sigma_{l_{2}}^{(j_{2},j_{0}/2)}(y)|^{2},$$

which yields

$$\varphi_{j_{1}+j_{2},j_{0}}\leq\frac{\log C_{5}}{2(j_{1}+j_{2})}+\frac{j_{1}}{j_{1}+j_{2}}\varphi_{j_{1},j_{0}/2}+\frac{j_{2}}{j_{1}+j_{2}}\varphi_{j_{2},j_{0}/2}.$$

Note finally that $\varphi_{j,j_{0}/2}\leq\varphi_{j,j_{0}}$ since $\varphi=\psi-h\leq 0$, therefore $\hat{\varphi}_{j}:=\varphi_{2^{j},j_{0}}+2^{-j-2}\log C_{5}$ is decreasing.

Step3. It remains to make $\hat{\varphi}_{j}$ smooth. Indeed it has all the other required properties: it is decreasing and by Step 1 we have for all $x\in X$,

(5)

$$\varphi(x)\leq\hat{\varphi}_{j}(x)\leq(1-j_{0}2^{-j})\sup_{B(x,r)}\varphi+\frac{C_{6}-n\log r}{2^{j}},$$

so that $\hat{\varphi}_{j}\rightarrow\varphi$. Let $\sigma_{1+s_{2^{j}}}^{(2^{j})},\ldots,\sigma_{N_{j}}^{(2^{j})}\in\Gamma(X,L^{2^{j}})$ be such that $(\sigma_{l}^{(2^{j})})_{l}$ is a basis of $\Gamma(X,L^{2^{j}})$ and set

$$\varphi_{j}:=\frac{1}{2^{j+1}}\log\left[\sum_{l=1}^{s_{2^{j}}}|\sigma_{l}^{(2^{j})}|^{2}+\varepsilon_{j}\sum_{l=1+s_{2^{j}}}^{N_{j}}|\sigma_{l}^{(2^{j})}|^{2}\right]+\frac{\log C_{5}}{2^{j+2}}-h.$$

Clearly $\varphi_{j}\in PSH(X,\omega)$. Moreover $\varphi_{j}\in{\mathcal{C}}^{\infty}(X)$ because $L^{2^{j}}$ is very ample if $j$ is large enough (hence we can find, for every $x\in X$, a holomorphic section of $L^{2^{j}}$ on $X$ which does not vanish at $x$). Finally we can choose $\varepsilon_{j}>0$ that decrease so fast to zero that $(\varphi_{j})$ is still decreasing and converges to $\varphi$. ∎