Proposition 2 1 the additive inverse is unique 2 for

This preview shows page 8 - 11 out of 15 pages.

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
Image of page 8

Subscribe to view the full document.

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
Image of page 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.
Image of page 10

Subscribe to view the full document.

Image of page 11
  • Fall '08
  • Staff
  • Math, Introductory Real Analysis, Dr. F. Dos Reis, Dr. F. Dos

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern