Least Upper Bound Property (LUBP)

• categories: Real Analysis, Definition

Definition 1

Say $E\subset S$ ordered if $\exists \beta :\forall x\in E⟹x\le \beta$ We we call $\beta$ an upper bound for $E$

Defintion 2

If $\exists \alpha \in S\mid \alpha$ is upper bound and $\nexists \beta \le \alpha \mid \beta$ is upper bound ⇒ $\alpha$ is LUB (sup)

Definition 3

LUB property. Every nonempty subset that has upper bound has LUB

Observation

Finite ordered sets have sup