Given $A= \bigcup_\limits{i \in \mathbb{N}} A_i $ and given a 'proposition' (voluntary vague) of which we want to evaluate the infimum, is it true that
$\displaystyle{\text{inf}_{A} \leq \sum_\limits{i \in \mathbb{N}}\text{inf}_{A_i}} \ \ \ \ \ \Large{?}$
I want to restrict to the case in which every infimum is at least $0$.
For example, the proposition can be a function, or also the interval-coverings of $A$, as in the Lebesgue measure (on which this proposition is true).