Proof.
In a pseudometric space, sequential completeness implies completeness for Cauchy nets. So for the first assertion, we may assume that $(\varphi _ j)_ j$ is a Cauchy sequence.
We first treat the case of a decreasing sequence $(\gamma _ j)_ j$ with $\int _ X\theta _{\gamma _ j}^ n\geq \delta $ for all $j$. Put $\tilde\gamma _ j=P_\theta [\gamma _ j]$. Then $(\tilde\gamma _ j)_ j$ is a decreasing sequence of model potentials, $d_ S(\gamma _ j,\tilde\gamma _ j)=0$, and
\[ \int _ X\theta _{\tilde\gamma _ j}^ n=\int _ X\theta _{\gamma _ j}^ n\geq \delta \]
by proposition 16.3.10. By proposition 16.4.4, the decreasing limit $\tilde\gamma =\inf _ j\tilde\gamma _ j$ is model and
\[ \lim _{j\to \infty }\int _ X\theta _{\tilde\gamma _ j}^ n=\int _ X\theta _{\tilde\gamma }^ n\geq \delta . \]
Hence $\tilde\gamma _ j\to \tilde\gamma $ in $d_ S$ by corollary 16.9.14, and therefore $\gamma _ j$ also converges in $d_ S$.
Next let $(\varphi _ j)_{j>0}$ be a $d_ S$-Cauchy sequence with masses bounded from below by $\delta $. Put
\[ \tilde{\varphi }_ j=P_\theta [\varphi _ j]. \]
By corollary 16.3.20, example 16.8.6, proposition 16.9.6, the $\tilde{\varphi }_ j$'s are model potentials, $d_ S(\varphi _ j,\tilde{\varphi }_ j)=0$, and
\[ \int _ X\theta _{\tilde{\varphi }_ j}^ n=\int _ X\theta _{\varphi _ j}^ n\geq \delta \]
by proposition 16.3.10. Replacing $\varphi _ j$ by $\tilde{\varphi }_ j$, we may therefore assume that the $\varphi _ j$'s are model potentials. Passing to a subsequence, we may also assume that
\[ d_ S(\varphi _ j,\varphi _{j+1})\leq C_ n^{-2j}, \]
where $C_ n$ is the constant in corollary 16.9.9. Define
\[ \psi _ j=\operatorname *{sup}_{k\geq j}{}\! {}^{*}\varphi _ k \]
for each $j>0$. As in the proof of proposition 16.9.17 Step 1, especially (16.9.20), we know that
\[ \lim _{j\to \infty } d_ S(\varphi _ j,\psi _ j)=0. \]
The sequence $(\psi _ j)_ j$ is decreasing and satisfies $\int _ X\theta _{\psi _ j}^ n\geq \delta $ by theorem 13.13.11. By the first paragraph, $(\psi _ j)_ j$ converges in $d_ S$. Hence the chosen subsequence of $(\varphi _ j)_ j$ converges, and the original Cauchy sequence converges as well.
Finally, suppose that $(\varphi _ j)_{j\in I}$ is a decreasing net with masses bounded from below by $\delta $. For each $k=0,\ldots ,n$, the net
\[ \int _ X\theta _{\varphi _ j}^ k\wedge \theta _{V_\theta }^{n-k} \]
is decreasing and bounded from below, hence Cauchy as a real-valued net. If $j'\geq j$, then $\varphi _{j'}\leq \varphi _ j$, and 16.9.7 expresses $d_ S(\varphi _{j'},\varphi _ j)$ as a finite sum of the corresponding differences of these mixed masses. For two sufficiently large indices, we choose a common upper bound in $I$ and use the triangle inequality. It follows that $(\varphi _ j)_{j\in I}$ is $d_ S$-Cauchy. The first assertion gives its convergence.
$\square$
Pome
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