Supremum and Infimum

Definition 1. 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

  \[\inf A.\]

Proposition 1. Let $X$ 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.