Passing to the infimum/Statement

Note: if you came here from a search engine, you should see Passing to the infimum as this is a subpage for transclusion.

Statement

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