Chapter 1 - The Real Numbers

# Chapter 1 - The Real Numbers - Chapter 1 The Real Numbers...

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Chapter 1 – The Real Numbers Section 1.1 – The Completeness Axiom: The Natural, Rational, and Irrational Numbers Pg 6, Definition – A set S of real numbers is said to be inductive provided that (i) the number 1 is in S and (ii) if the number x is in S, the number x+1 is also in S. Pg 7, Each rational number z may be expressed as z = m/n, where m and n are integers and either m or n is odd. Pg 7, If q is an integer and q 2 is even, then q is even. Pg 7, Proposition 1.1 – There is no rational number whose square equals 2. Pg 8, Definition – A nonempty set S of real numbers is said to be bounded above provided that there is a number c having the property that for all x in S. Such a number c is called an upper bound for S. Pg 8, The Completeness Axiom – Suppose that S is a nonempty set of real numbers that is bounded above. Then among the set of upper bounds for S there is a smallest, or least, upper bound ( l.u.b. S ) or (supremum of S denoted sup S ). Pg 8, Proposition 1.2 – Suppose that S is a nonempty set of real numbers that is bounded above, and that the number

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## This note was uploaded on 11/10/2011 for the course MATH 511 taught by Professor Higdon during the Spring '11 term at Oregon State.

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Chapter 1 - The Real Numbers - Chapter 1 The Real Numbers...

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