SalasSV_10_01 - CHAPTER 10 10.1 THE LEAST UPPER BOUND AXIOM...

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CHAPTER SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS 10 ± 10.1 THE LEAST UPPER BOUND AXIOM So far our approach to the real number system has been somewhat primitive. We have simply taken the point of view that there is a one-to-one correspondence between the set of points on a line and the set of real numbers, and that this enables us to measure all distances, take all roots of nonnegative numbers, and, in short, Fll in all the gaps left by the set of rational numbers. This point of view is basically correct and has served us well, but it is not sufFciently sharp to put our theorems on a sound basis, nor is it sufFciently sharp for the work that lies ahead. We begin with a nonempty set S of real numbers. As indicated in Section 1.2, a number M is an upper bound for S if x M for all x S . It follows that if M is an upper bound for S , then every number in [ M , ) is also an upper bound for S . Of course, not all sets of real numbers have upper bounds. Those that do are said to be bounded above . It is clear that every set that has a largest element has an upper bound: if b is the largest element of S , then x b for all x S . This makes b an upper bound for S . The converse is false: the sets S 1 = ( −∞ , 0) and S 2 = ± 1 2 , 2 3 , 3 4 , ... , n n + 1 , ... ² both have upper bounds (for instance, 2 is an upper bound for each set), but neither has a largest element. Let’s return to the Frst set, S 1 . While ( −∞ , 0) does not have a largest element, the set of its upper bounds, namely [0, ), does have a smallest element, namely 0. We call 0 the least upper bound of ( −∞ , 0). Now let’s reexamine S 2 . While the set of quotients n n + 1 = 1 1 n + 1 , n = 1, 2, 3, ... , 585
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586 ± CHAPTER 10 SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS does not have a greatest element, the set of its upper bounds, [1, ), does have a least element, 1. The number 1 is the least upper bound of that set of quotients. In general, if S is a nonempty set of numbers which is bounded above, then the least upper bound of S is an upper bound that is less than or equal to any other upper bound for S . We now state explicitly one of the key assumptions that we make about the real number system. It is called the least upper bound axiom , and it provides the sharpness and the clarity that we require. AXIOM 10.1.1 LEAST UPPER BOUND AXIOM Every nonempty set of real numbers that has an upper bound has a least upper bound. Some Fnd this axiom obvious; some Fnd it unintelligible. ±or those of you who Fnd it obvious, note that the axiom is not satisFed by the rational number system; namely, it is not true that every nonempty set of rational numbers that has a rational upper bound has a least rational upper bound. (±or a detailed illustration of this, we refer you to Exercise 33). Those who Fnd the axiom unintelligible will come to understand it by working with it. We indicate the least upper bound of a set
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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SalasSV_10_01 - CHAPTER 10 10.1 THE LEAST UPPER BOUND AXIOM...

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