PomeProposition 16.9.44.362688C3DDC9 Let $(\varphi _ i)_{i\in I}$, $(\psi _ i)_{i\in I}$ be nets in $\mathrm{PSH}(X,\theta )$ such that $\varphi _ i\xrightarrow {d_ S} \varphi \in \mathrm{PSH}(X,\theta )$ and $\psi _ i\xrightarrow {d_ S}\psi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi _ i\preceq _ P \psi _ i$ for all $i\in I$. Then $\varphi \preceq _ P \psi $.
Proposition 16.9.44362688C3DDC9
Proof. It follows from proposition 16.9.42 that
\[ \varphi _ i\lor \psi _ i\xrightarrow {d_ S}\varphi \lor \psi . \]
By 16.8.13, we have $\varphi _ i\lor \psi _ i\sim _ P \psi _ i$ for all $i\in I$. In particular, by proposition 16.9.6,
\[ \varphi _ i\lor \psi _ i\xrightarrow {d_ S}\psi . \]
By proposition 16.9.6 again, $\varphi \lor \psi \sim _ P\psi $ and hence $\varphi \preceq _ P \psi $ by 16.8.13. $\square$
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