Infimum

Infimum

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 } = 2^1/3

inf { (-1)ⁿ + 1/n | n = 1, 2, 3, ... } = -1

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

The infimum and supremum of S are related via

inf(S) = - sup(-S).

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're done right away).

Generalization

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

A infimum or greatest lower bound of S is an element l in P such that
- l <= x for all x in S, and
- for any p in P such that p <= x for all x in S it holds that p <= l.

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.

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.

Greatest lower bound property

See the article on the least upper bound property.