For $m\gg 1$, $m\mathcal{L}|_{X}$ is globally generated, which shows that the ideal sheaf $\mathfrak{a}_{m}$ is vertical. Since $\mathcal{O}_{\mathcal{X}}(m\mathcal{L})\otimes\mathfrak{a}_{m}$ is globally generated by the definition of $\mathfrak{a}_{m}$, it follows that $\varphi_{m}\in\mathcal{D}(X)$ is $\theta$-psh by Lemma 5.6. Note that $\mathfrak{a}_{m}\cdot\mathfrak{a}_{l}\subset\mathfrak{a}_{m+l}$ for all $m,l$. This yields the super-additivity property $m\varphi_{m}+l\varphi_{l}\leq(m+l)\varphi_{m+l}$. As a consequence, the pointwise limit $\lim_{m}\varphi_{m}$ exists and coincides with $\sup_{m}\varphi_{m}$.

Step 1. Let us first prove that $P_{\theta}(0)=\sup_{m}\varphi_{m}$ on $X^{\mathrm{qm}}$. This is similar to Step 2 of Theorem 5.11. Since $\varphi_{m}$ is $\theta$-psh and $\varphi_{m}\leq 0$ for all $m$, we have $\sup_{m}\varphi_{m}\leq P_{\theta}(0)$ on $X$. To see that equality holds on $X^{\mathrm{qm}}$, pick $\varepsilon>0$ and $x\in\emb_{\mathcal{X}^{\prime}}(\Delta_{\mathcal{X}^{\prime}})$ for some SNC model $\mathcal{X}^{\prime}$ dominating $\mathcal{X}$. By Lemma 8.4 there exists a $\theta$-psh model function $\varphi$ such that $\varphi\leq 0$ and $\varphi(x)\geq P_{\theta}(0)(x)-\varepsilon$. Replacing $\mathcal{X}^{\prime}$ by a higher model, we may assume that $\varphi=\varphi_{D}$ is determined by some divisor $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$. Invoking Proposition 5.2 we may also assume that there exists $D^{\prime}\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ with $-\varepsilon\leq\varphi_{D^{\prime}}\leq 0$ on $X$ and $\pi^{*}\mathcal{L}+D+D^{\prime}$ ample. Since $D+D^{\prime}\leq 0$ we then have

$$\mathcal{O}_{\mathcal{X}^{\prime}}(m\pi^{*}\mathcal{L}+m(D+D^{\prime}))\subset\mathcal{O}_{\mathcal{X}^{\prime}}(m\pi^{*}\mathcal{L}).$$

Now the left-hand side is globally generated for some $m$, and we conclude that

$$\mathcal{O}_{\mathcal{X}^{\prime}}(m(D+D^{\prime}))\subset\mathcal{O}_{\mathcal{X}^{\prime}}\cdot\mathfrak{a}_{m},$$

hence

$$P_{\theta}(0)(x)\leq\varphi_{D}(x)+\varepsilon\leq\varphi_{D+D^{\prime}}(x)+2\varepsilon\leq\frac{1}{m}\log|\mathfrak{a}_{m}|(x)+2\varepsilon\leq\sup_{l}\varphi_{l}(x)+2\varepsilon.$$

Step 2. Introduce for each $m\in\mathbf{N}$ the asymptotic multiplier ideal $\mathfrak{b}_{m}=\mathcal{J}(\mathfrak{a}_{\bullet}^{m})\subset\mathcal{O}_{\mathcal{X}}$ associated to the graded sequence $\mathfrak{a}_{\bullet}$. We refer to Appendix B for the definition and the proof of the fundamental properties of multiplier ideals in our present setting. We shall use the following results. First we have the elementary inclusion $\mathfrak{a}_{m}\subset\mathfrak{b}_{m}$ for all $m$. Second, the subadditivity property (cf. Theorem B.7) implies $\mathfrak{b}_{ml}\subset\mathfrak{b}^{l}_{m}$ for any $l,m$. We infer that $\mathfrak{a}_{ml}\subset\mathfrak{b}_{ml}\subset\mathfrak{b}_{m}^{l}$ for any $m,l$ and hence

(8.1)

$$\tfrac{1}{m}\log|\mathfrak{b}_{m}|\geq\sup_{l}\tfrac{1}{ml}\log|\mathfrak{a}_{ml}|=\sup_{l}\varphi_{ml}=P_{\theta}(0)$$

on $X^{\mathrm{qm}}$ for all $m$, where the last equality follows from the first step.

Since both $P_{\theta}(0)$ and $\varphi_{m}$ remain unchanged when $\mathcal{X}$ is replaced with a higher model, we may assume that there exists an effective divisor $E\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ such that $\mathcal{A}:=\mathcal{L}-E$ is ample on $\mathcal{X}$. By the uniform global generation property of multiplier ideals (Theorem B.8) we may then choose $m_{0}\in\mathbf{N}$ such that $\mathcal{O}_{\mathcal{X}}(m\mathcal{L}+m_{0}\mathcal{A})\otimes\mathfrak{b}_{m}$ is globally generated for all $m$. Since $\mathcal{O}_{\mathcal{X}}(m\mathcal{L}+m_{0}\mathcal{A})$ injects in $\mathcal{O}_{\mathcal{X}}((m+m_{0})\mathcal{L})$ by multiplying with the canonical section of $\mathcal{O}_{\mathcal{X}}(m_{0}E)$, it follows that

$$\log|\mathfrak{b}_{m}|\leq\log|\mathfrak{a}_{m+m_{0}}|+m_{0}\varphi_{E}.$$

Replacing $m$ with $m-m_{0}$ and using (8.1) we infer $(m-m_{0})P_{\theta}(0)\leq m\varphi_{m}+m_{0}\varphi_{E}$, so that

$$\varphi_{m}\leq P_{\theta}(0)\leq\tfrac{m}{m-m_{0}}\varphi_{m}+\tfrac{m_{0}}{m-m_{0}}\varphi_{E}$$

on $X^{\mathrm{qm}}$ for $m\gg 1$. As $\varphi_{m}$, $P_{\theta}(0)$ and $\varphi_{E}$ are all $\theta$-psh, Proposition 7.6 shows that this inquality extends to all of $X$. Now $\varphi_{E}$ is bounded and $\varphi_{m}$ is uniformly bounded, as follows from $\varphi_{1}\leq\varphi_{m}\leq 0$, so $\varphi_{m}$ converges uniformly on $X$ to $P_{\theta}(0)$, as was to be shown. ∎