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**|*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*).

*S*) ≥

*A*, one only has to show that

*x*≥

*A*for all

*x*in

*S*. Showing that inf(

*S*) ≤

*A*is a bit harder: for any ε > 0, you have to exhibit an element

*x*in

*S*with

*x*≤

*A*+ ε.

[ *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.

### Generalization

One can define infima for subsets *S* of arbitrary partially ordered sets (*P*, <=) as follows:

*S*has a infimum, then the infimum is unique: if

*l*

_{1}and

*l*

_{2}are both infima of

*S*then it follows that

*l*

_{1}<=

*l*

_{2}and

*l*

_{2}<=

*l*

_{1}, and since <= is antisymmetric it follows that

*l*

_{1}=

*l*

_{2}.

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.