I) If a nonempty set of real numbers is bounded above, then it has a supremum. Property (I) is ca... more I) If a nonempty set of real numbers is bounded above, then it has a supremum. Property (I) is called completeness, and we say that the real number system is a complete ordered field. It can be shown that the real number system is essentially the only complete ordered field; that is, if an alien from another planet were to construct a mathematical system with properties (A)-(I), the alien's system would differ from the real number system only in that the alien might use different symbols for the real numbers and C, , and <. Theorem 1.1.3 If a nonempty set S of real numbers is bounded above; then sup S is the unique real numberˇsuch that (a) x ġfor all x in S I (b) if > 0 .no matter how small/; there is an x 0 in S such that x 0 >ˇ : Section 1.1 The Real Number System 5
I) If a nonempty set of real numbers is bounded above, then it has a supremum. Property (I) is ca... more I) If a nonempty set of real numbers is bounded above, then it has a supremum. Property (I) is called completeness, and we say that the real number system is a complete ordered field. It can be shown that the real number system is essentially the only complete ordered field; that is, if an alien from another planet were to construct a mathematical system with properties (A)-(I), the alien's system would differ from the real number system only in that the alien might use different symbols for the real numbers and C, , and <. Theorem 1.1.3 If a nonempty set S of real numbers is bounded above; then sup S is the unique real numberˇsuch that (a) x ġfor all x in S I (b) if > 0 .no matter how small/; there is an x 0 in S such that x 0 >ˇ : Section 1.1 The Real Number System 5
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