Upper and lower bounds

Characteristics of sets on the real line. The least upper bound of a given set of real numbers is the smallest number bounding this set from above; its greatest lower bound is the largest number bounding it from below. This will now be restated in more detail. Let there be given a subset of the real numbers. A number is said to be its least upper bound, denoted by (from the Latin "supremum" — largest), if every number satisfies the inequality , and if for any there exists an such that . A number is said to be the greatest lower bound of , denoted by (from the Latin "infimum" — smallest), if every satisfies the inequality , and if for any there exists an such that .

Examples.

if the set consists of two points and , , then

These examples show, in particular, that the least upper bound (greatest lower bound) may either belong to the set (e.g. in the case of the interval ) or not belong to it (e.g. in the case of the interval ). If a set has a largest (smallest) member, this number will clearly be the least upper bound (greatest lower bound) of the set.

The least upper bound (greatest lower bound) of a set not bounded from above (from below) is denoted by the symbol (respectively, by the symbol ). If is the set of natural numbers, then

If is the set of all integers, both positive and negative, then

Each non-empty set of real numbers has a unique least upper bound (greatest lower bound), finite or infinite. All non-empty sets bounded from above have finite least upper bounds, while all those bounded from below have finite greatest lower bounds.

The terms least upper (greatest lower) limit of a set are also sometimes used instead of the least upper bound (greatest lower bound) of a set, in one of the senses defined above. By the least upper bound (greatest lower bound) of a real-valued function, in particular of a sequence of real numbers, one means the least upper bound (greatest lower bound) of the set of its values (cf. also Upper and lower limits).

References

[1]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[2]	L.D. Kudryavtsev, "A course in mathematical analysis" , 1 , Moscow (1988) (In Russian)
[3]	S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)

Comments

Commonly, an upper bound of a set of real numbers is a number such that for all one has . The least upper bound of is then defined as an upper bound such that for every upper bound one has .

Analogous definitions hold for a lower bound and the greatest lower bound. If the least upper bound of belongs to , then it is called the maximum of .

If the greatest lower bound of belongs to , then it is called the minimum of .

References

[a1]	T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)
[a2]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
[a3]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)