Supremum and Infimum

DEF-SUP. Supremum and Infimum.

Let $(X,\preceq)$ be a partially ordered set and $A\subseteq X$.

• If the least upper bound of $A$ exists, it is called the supremum of $A$ and denoted by $\sup A$.

• If the greatest lower bound of $A$ exists, it is called the infimum of $A$ and denoted by $\inf A$.

PROP-SUP-MAX.

Let $(X,\preceq)$ be a partially ordered set and $A\subseteq X$. If $\sup A$ exists, the following statements are equivalent:

1. $\sup A\in A$.
2. $\sup A=\max A$.
3. $\max A$ exists.

If $\inf A$ exists, the following statements are equivalent:

4. $\inf A\in A$.
5. $\inf A=\min A$.
6. $\min A$ exists.