In analysis the infimum or greatest lower bound of a set 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:
- inf { x in R | 0 < x < 1 } = 0
- inf { x in R | x3 > 2 } = 21/3
- inf { (-1)n + 1/n | n = 1, 2, 3, ... } = -1
The infimum and supremum of S are related via
- inf(S) = - sup(-S).
[ Actually the last sentence above is technically not true, since it is sufficient to show there exists an x in S such that x ≤ A. For example you don't need epsilons to see that inf(set of positive integers) ≤ 100, because 9 is in the set and 9 < 100. If that fails, then use the strategy above. ]
See also: limit inferior.
One can define infima for subsets S of arbitrary partially ordered sets (P, <=) as follows:
In an arbitrary partially ordered set, there may exist subsets which don't have a infimum.
In a lattice every nonempty finite subset has an infimum, and in a complete lattice every subset has an infimum.
See the article on the least upper bound property.Generalization
It can easily be shown that, if S has a infimum, then the infimum is unique: if l1 and l2 are both infima of S then it follows that l1 <= l2 and l2 <= l1, and since <= is antisymmetric it follows that l1 = l2.Greatest lower bound property