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# Proposition 2 1 the additive inverse is unique 2 for

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Proposition 2. 1. The additive inverse is unique. 2. For any real number a , 0 * a = 0. 3. ( - 1)( - 1) = 1. 4. The additive inverse of a + b is ( - a ) + ( - b ). 5. for any real number a , - a = ( - 1) * a . Axiom 2. R is totally ordered: There is a relation > that satisfies 1. If a > 0 and b > 0, then a + b > 0 2. If a > 0 and b > 0, then a * b > 0 3. For each a only one of the following is true (a) a > 0, (b) a = 0, (c) - a > 0 Definition 3. We say that a < b if b - a > 0 Proposition 4. Given three real numbers a , b , c , 1. 0 < a 2 if a 6 = 0 2. If a < b and b < c then a < c 3. If a < b and 0 < c , then ac < bc 4. If a < b , for any c , a + c < b + c 8

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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Proposition 5. 1. If a is a positive real number, then its multiplicative inverse a - 1 is positive. 2. For any real numbers such that 0 < a < b , then b - 1 < a - 1 . Definition 6. The absolute value of a , written | a | is • | a | = a if a > 0 • | a | = - a if a < 0 • | a | = 0 if a = 0 Proposition 7. For any a , b in R , • | a * b | = | a | * | b | • | a + b | 6 | a | + | b | • || a | - | b || 6 | a - b | Proposition 8. For any real numbers such that a < b < c , | b | 6 max ( | a | , | c | ). Axiom 3. (Axiom of continuity) Suppose that all real numbers are sepa- rated into two collections which we denote by L and R , in such a way that 1. Every number is either in L or in R . 2. Each collection contains at least 1 element. 3. If a is in L and b is in R , then a < b . then there is a number c in R such that all numbers less than c are in L and all numbers greater that c are in R . Proposition 9. The cut number c is unique. Theorem 10. (Archimedian law of real numbers) Let a and b be 2 positive real numbers. There exists a positive integer n such that b < na . Proposition 11. If a real number y satisfies that 0 6 y 6 1 n for any natural number n , then y = 0. Proposition 12. For any positive rel number y , there exists a real number c such that c 2 = y . Theorem 13. Any system that satisfies Axiom 1, 2, and 3 is isomorphic to R 2 Axioms of N (2.3) Axiom 4. The natural numbers are the smallest class of real numbers that satisfies 1. 1 is an element of N 2. If n is an element of N , n + 1 is an element of N Proposition 14. Any natural number either equals to 1 or is greater than 1. 9
Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Proposition 15. For any 2 natural numbers p < n , there exist a natural number m such that n = p + m . Proposition 16. Let n be a natural number. there is no natural number between n and n + 1. Proposition 17. If S is a class of positive integers containing at least 1 element, it contains a smallest element. 3 Rational & irrational numbers (2.5) Definition 18. A rational number is a number that can be written in the form ± p q , where p and q are 2 natural numbers. An irrational number is a number that is not rational Theorem 19. The number 2 is irrational. Theorem 20. Given to numbers a < b , there is a rational number between a and b . There is an irrational number between a and b . 4 Least upper bounds, greatest lower bounds (2.6) Theorem 21. If S is a set of real numbers which is not empty and which has an upper bound, then it has a least upper bound.

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• Fall '08
• Staff
• Math, Introductory Real Analysis, Dr. F. Dos Reis, Dr. F. Dos

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