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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.

Examples:

inf { x in R | 0 < x < 1 } = 0
inf { x in R | x3 > 2 } = 21/3
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 xA 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 xA + 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:

  • 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.



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