Infimum

analysis set-theory

Definition

Infimum

The infimum (or greatest lower bound) of a set is the largest real number that is less than or equal to every element in . It is denoted by .

Formally, if:

1. is a lower bound: .
2. is the greatest such bound: if is any lower bound of , then .

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

Relation to Minimum

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

Properties

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