Infimum

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A closely related concept is the supremum, which is the smallest upper bound rather than the greatest lower bound.

Definition

An infimum or greatest lower bound (AKA: g.l.b) of a subset [ilmath]A\subseteq X[/ilmath] of a poset [ilmath](X,\preceq)[/ilmath]:

• [ilmath]\text{inf}(A)[/ilmath]

such that:

1. [ilmath]\forall a\in A[\text{inf}(A)\le a][/ilmath] (that [ilmath]\text{inf}(A)[/ilmath] is a lower bound)
2. [ilmath]\forall x\in\underbrace{\{y\in X\ \vert\ \forall a\in A[y\le a]\} }_{\text{The set of all lower bounds} }\ \ [\text{inf}(A)\ge x][/ilmath] (that [ilmath]\text{inf}(A)[/ilmath] is an upper bound of all lower bounds of [ilmath]A[/ilmath])
   • Claim 1: , this is the same as [ilmath]\forall x\in X\exists a\in A[x>\text{inf}(A)\implies a<x][/ilmath]

Proof of claims

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See also

• Passing to the infimum
• Supremum
  • Passing to the supremum