Supremum

analysis set-theory

Definition

Supremum

The supremum (or least upper bound) of a set is the smallest real number that is greater than or equal to every element in . It is denoted by .

Formally, if:

1. is an upper bound: .
2. is the least such bound: if is any upper bound of , then .

If the set is not bounded above, we define . For the empty set, .

Relation to Maximum

A supremum is called a maximum if it is an element of the set . While every non-empty set that is bounded above has a supremum (see Completeness of the Real Numbers), it does not necessarily have a maximum (e.g., the open interval has a supremum of but no maximum).

Properties

• Uniqueness: If a supremum exists, it is unique.
• Approximation Property: For any , there exists an element such that .
• Sequences: The supremum of a sequence is the supremum of the set of its values .