Passing to the infimum/Statement

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Let [ilmath]A,B\subseteq X[/ilmath] be subsets of [ilmath]X[/ilmath] where [ilmath](X,\preceq)[/ilmath] is a poset. Then:

  • If [ilmath]\forall a\in A\exists b\in B[b\preceq a][/ilmath] then [ilmath]\text{inf}(B)\preceq\text{inf}(A)[/ilmath] (provided both infima exist)
Note: See Passing to the infimum for a discussion of the motivation and conditions