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 $A,B\subseteq X$ be subsets of $X$ where $(X,\preceq)$ is a poset. Then:

• If $\forall a\in A\exists b\in B[b\preceq a]$ then $\text{inf}(B)\preceq\text{inf}(A)$ (provided both infima exist)

Note: See Passing to the infimum for a discussion of the motivation and conditions