SUP#DEF. Supremum and Infimum.
Let $X$ 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 $\inf A$.
Let $X$ be a partially ordered set and $A\subseteq X$. If $\sup A$ exists, the following statements are equivalent:
- \[\sup A \in A.\]
- \[\sup A = \max A.\]
$\max A$ exists.
If $\inf A$ exists, the following statements are equivalent:
- \[\inf A\in A.\]
- \[\inf A=\min A.\]
$\min A$ exists.