PomeProposition 22.1.45.C8734FD15163 Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.42). Let $C\in \mathbb {R}$. Then
\[ \inf _{i\in I} (\Gamma ^ i+C)=\inf _{i\in I} \Gamma ^ i+C. \]
Suppose that $(\Gamma '^ i)_{i\in I}$ is another decreasing net in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta ')$ satisfying (22.1.42). Suppose that $\Gamma ^ i\leq \Gamma '^ i$ for all $i\in I$. Then
\[ \inf _{i\in I}\Gamma ^ i\leq \inf _{i\in I}\Gamma '^ i. \]
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