Theorem 16.9.5589B76134A3A3

Proof. First observe that since $d_ S$ is a pseudometric, it suffices to prove (16.9.56) when $I=\mathbb {Z}_{>0}$ as partially ordered sets.

By corollary 16.9.36, after adding a Kähler form to $\theta $, we may assume that the masses of $\varphi _ j$ and of $\varphi $ are bounded from below by a positive constant.

By theorem 16.9.49, we may assume that $\varphi _ j$ and $\varphi $ are both $\mathcal{I}$-model and hence model. When proving (16.9.56), we are free to pass to subsequences.

By proposition 16.9.17 and a sandwich argument, we may assume that the sequence $(\varphi _ j)_ j$ is either increasing or decreasing. In the increasing case, the Lelong numbers $\nu (\varphi _ j,E)$ decrease to a limit at least $\nu (\varphi ,E)$, while $L^1$-convergence gives the opposite inequality by upper semicontinuity of Lelong numbers. In the decreasing case, (16.9.56) follows from proposition 16.4.4. $\square$