Bounds

Overview

Suppose $S$ is a nonempty set of real numbers and suppose there are numbers $L$ and $U$ such that $L\le x\le U$ for all $x\in S$. Then $S$ is said to be bounded below by $L$ and bounded above by $U$. The number $L$ is said to be a lower bound for $S$; the number $U$ is said to be an upper bound for $S$.

If $L\in S$, then $L$ is the minimum element of $S$ (denoted $L=\min S$). Likewise, if $U\in S$, then $U$ is the maximum element of $S$ (denoted $U=\max S$). A set with no lower bound is said to be unbounded below. A set with no upper bound is said to be unbounded above.

Least Upper Bounds

A number $B$ is called a least upper bound (or supremum) of a nonempty set $S$ if $B$ is an upper bound for $S$ and no number less than $B$ is an upper bound for $S$. This is denoted as $B=\mathrm{lub}S$ or $B=\sup S$.

Completeness Axiom

Every nonempty set $S$ of real numbers which is bounded above has a supremum; that is, there is a real number $B$ such that $B=\sup S$.

Greatest Lower Bounds

A number $B$ is called a greatest lower bound (or infimum) of a nonempty set $S$ if $B$ is a lower bound for $S$ and no number greater than $B$ is a lower bound for $S$. This is denoted as $B=\mathrm{glb}S$ or $B=\inf S$.

Bibliography

• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).