Infimum: Encyclopedia II - Infimum - Infima of real numbers

Infimum - Infima of real numbers

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:

Note that the infimum does not have to belong to the set (as in these examples). If the infimum value belongs to the set then we can say there is a smallest element in the set.

The notions of infimum and supremum are dual in the sense that

In general, in order to show that inf(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 (of course, if you can find an element x in S with x ≤ A, you are done right away).

See also: limit inferior.

Adapted from the Wikipedia article "Infima of real numbers", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki