Introduction to the Mathematical and Statistical Foundations of Econometrics

Supremum and Infimum

The supremum of a sequence of real numbers, or a real function, is akin to the notion of a maximum value. In the latter case the maximum value is taken at some element of the sequence, or in the function case some value of the argument. Take for example the sequence an = (— 1)n/n for n = 1, 2,..., that

is, ai = —1, a2 = 1/2, a3 = -1/3, a4 = 1/4,________ Then clearly the maximum

value is 1/2, which is taken by a2. The latter is what distinguishes a maximum from a supremum. For example, the sequence an = 1 — 1/n for n = 1, 2,... is bounded by 1: an < 1 for all indices n > 1, and the upper bound 1 is the lowest possible upper bound; however, a finite index n for which an = 1 does not exist. More formally, the (finite) supremum of a sequence an(n = 1, 2, 3,...) is a number b denoted by supn> an such that an < b for all indices n > 1, and for every arbitrary small positive number є there exists a finite index n such that an > b — є. Clearly, this definition fits a maximum as well: a maximum is a supremum, but a supremum is not always a maximum.

If the sequence an is unbounded from above in the sense that for every arbitrary, large real number M there exists an index n > 1 for which an > M, then we say that the supremum is infinite: supn> an =<x>.

The notion of a supremum also applies to functions. For example, the function f (x) = exp(—x2) takes its maximum 1 at x = 0, but the function f (x) = 1 — exp(—x2) does not have a maximum; it has supremum 1 because f (x) < 1 for all x, but there does not exist a finite x for which f (x) = 1. As another example, let f (x) = x on the interval [a, b]. Then b is the maximum of f (x)on [a, b],but b is only the supremum f (x) on [a, b) because b is not contained in [a, b). More generally, the finite supremum of a real function f (x) on a set A, denoted by supxeA f (x), is a real number b such that f (x) < b for all x in A, and for every arbitrary, small positive number є there exists an x in A such that f (x) > b — є. If f (x) = b for some x in A, then the supremum coincides with the maximum. Moreover, the supremum involved is infinite, supxeA f (x) = ro, if for every arbitrary large real number M there exists an x in A for which f(x ) > M.

The minimum versus infimum cases are similar:

inf an = — sup(—an) and infx ea f (x) = — sup^A (—f (x)).

n>1 n>1

The concepts of supremum and infimum apply to any collection {ca, a e A} of real numbers, where the index set A may be uncountable, for we may interpret ca as a real function on the index set A - for instance, ca = f (a).