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We now introduce the notions of upper bound and lower bound for a set of real numbers. These ideas will be of utmost importance in later sections.
Let S be a nonempty subset of R
(a) The set S is said to be bounded above if there exists a number u element R such that s last than or equal u for all s element S. Each such number u is called an upper bound of S; (b) The set S is said to be bounded below if there exists a number w element R such that w last than or equal s for all s elemen S. Each such number w is called a lower bound of S; (c) A set is said to be bounded if it is bounded above and bounded below. A set is said to be unbounded if it is not bounded
From the Trichotomy Property, we are assured that if a element R and a not as 0, then exactly one of the numbers a and -a is positive. The absolute value of a not as 0 is defined to be the positive one of these two numbers. The ansolute value of 0 is defined to be 0.
The "order properties" of R refer to thr notions of positivity and inequalities between real numbers. As with the algebraic structure of thr system of real numbers, we proceed by isolating three basic properties from which all other properties and calculations with inequalities can be deduced. The simplest way to do this to identify a special subset of Rby using the notion of "positivity"
In this chapter we will discuss the essential properties of the real number system R. Although it is possible to give a formal sonctruction of this system on the basis of a more primitive set (such as the set N of natural numbers or the set Q of rational numbers), we have chosen not to do so. Instead, we exhibit a list of fundamental properties associated with the real numbers and show how further properties can be deduced from them. This kind of activity is much more useful in learning the tools of analysis than examining the logical difficultis of contructing a model for R