Infima of Real Numbers
In analysis the infimum or greatest lower bound of a subset S of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists (because S is not bounded below), then we define inf(S) = −∞. If S is empty, we define inf(S) = ∞ (see extended real number line).
An important property of the real numbers is that every set of real numbers has an infimum (any bounded nonempty subset of the real numbers has an infimum in the non-extended real numbers).
Examples:
If a set has a smallest element, as in the first example, then the smallest element is the infimum for the set. (If the infimum is contained in the set, then it is also known as the minimum). As the last three examples show, the infimum of a set does not have to belong to the set.
The notions of infimum and supremum are dual in the sense that
- ,
where
See also: limit inferior.
Read more about this topic: Infimum
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