Upper and lower bounds

2020 Mathematics Subject Classification: Primary: 26A03 Secondary: 06A [MSN][ZBL]

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 $X$ of the real numbers. A number $\beta$ is said to be its least upper bound, denoted by $\sup X$ (from the Latin "supremum" — largest), if every number $x\in X$ satisfies the inequality $x\leq\beta$, and if for any $\beta'<\beta$ there exists an $x'\in X$ such that $x'>\beta'$. A number $\alpha$ is said to be the greatest lower bound of $X$, denoted by $\inf X$ (from the Latin "infimum" — smallest), if every $x\in X$ satisfies the inequality $x\geq\alpha$, and if for any $\alpha'>\alpha$ there exists an $x'\in X$ such that $x'<\alpha'$.

Examples.

$$\inf[a,b]=a,\quad\sup[a,b]=b;$$

$$\inf(a,b)=a,\quad\sup(a,b)=b;$$

if the set $X$ consists of two points $a$ and $b$, $a<b$, then

$$\inf X=a,\quad\sup X=b.$$

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 $[a,b]$) or not belong to it (e.g. in the case of the interval $(a,b)$). 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 $+\infty$ (respectively, by the symbol $-\infty$). If $\mathbf N=\{1,2,\dots\}$ is the set of natural numbers, then

$$\inf\mathbf N=1,\quad\sup\mathbf N=+\infty.$$

If $\mathbf Z$ is the set of all integers, both positive and negative, then

$$\inf\mathbf Z=-\infty,\quad\sup\mathbf Z=+\infty.$$

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 $S$ of real numbers is a number $b$ such that for all $x\in S$ one has $x\leq b$. The least upper bound of $S$ is then defined as an upper bound $B$ such that for every upper bound $b$ one has $B\leq b$.

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

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

The fundamental axiom of the real number system, or continuity axiom, may be expressed in the form Every non-empty set of real numbers bounded above has a real number supremum.

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)