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 $A\subseteq X$ of a poset $(X,\preceq)$:

• $\text{inf}(A)$

such that:

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

Proof of claims

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See also

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