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**|*x*^{3}> 2 } = 2^{1/3} - 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.