Bounds, Upper and Lower

(in mathematics), important characteristics of sets on a number line.

The upper bound of a set E of real numbers is the smallest number of all numbers A, which possess the property such that for any x of E the inequality x ≤ A is satisfied. In other words, the upper bound of set E is that number a such that for any x of E the inequality x ≤ A is satisfied and that for any a′ < a a number x₀ of E will be found for which x₀ < a′. In this definition it is assumed that set E is not empty. For the existence of an upper bound it is necessary and sufficient that set E be bounded from above—that is, that numbers A exist such that x ≤ A for any x of E. This proposition is one of the forms of the principle of continuity of a number line (the so-called Weierstrass principle of continuity). If among the numbers of set E there is one greater than any of the others, then it is the upper bound of E. If, however, there is no such greatest number among the numbers of E, this set may still have an upper bound. For example, the upper bound of all negative numbers is equal to zero. The set of all positive numbers is not bounded from above and therefore has no upper bound; it is sometimes said that its upper bound is equal to + ∞.

The lower bound of set E is defined analogously to the upper bound as the greatest of the numbers B, which possess the property such that for any x of E the inequality x ≥ B is satisfied. The upper bound of set E is designated sup E (from the Latin supremum—“the highest”); the lower bound is designated inf E (from the Latin infimum—“the lowest”). The importance of the concepts of upper and lower bounds was explained by the German mathematician K. Weier-strass; they are basic for the rigorous exposition of the fundamentals of mathematical analysis. Analogous to the concept of upper and lower bounds for sets of numbers has been the introduction of the concepts of upper and lower bounds for any partially ordered sets.