PomeProposition 16.5.53.322796EE46DE Let $(\varphi _ i)_{i\in I}$ be a uniformly bounded from above non-empty family in $\mathcal{E}(X,\theta ;\phi )$ (resp. $\mathcal{E}^1(X,\theta ;\phi )$, $\mathcal{E}^{\infty }(X,\theta ;\phi )$), then so is $\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ i$.
Proposition 16.5.53322796EE46DE
Proof. Thanks to proposition 16.5.52, it suffices to show that
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ i\preceq \phi . \]
Since $\phi $ is model and $\varphi _ i\preceq \phi $, we know that
\[ \varphi _ i-\sup _ X \varphi _ i \leq \phi \]
for any $i\in I$. Since $(\varphi _ i)_{i\in I}$ is uniformly bounded from above by assumption, our assertion follows. $\square$
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