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:
The infimum and supremum of S are related via
[ 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