New PDF release: A First Course in Real Analysis

New PDF release: A First Course in Real Analysis

By M. H. Protter C. B. Morrey Jr.

ISBN-10: 1461599903

ISBN-13: 9781461599906

ISBN-10: 146159992X

ISBN-13: 9781461599920

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Extra info for A First Course in Real Analysis

Example text

Am+n is any finite sequence, m+n La+ L j i=1 i=m+1 aj ( m ) (m+n II aj • II aj ) • i=1 i=m+\ Intuitively, we know that the set of natural numbers N is identical with the set of positive integers. The fact that a natural number is positive follows from the Axiom of inequality and the definition of N. The proof of the next theorem is left to the reader. 26. Every natural number is positive. The next three theorems show that the natural numbers are ordered, that they are closed under addition (that is, they satisfy Axiom A-I), and that they are closed under multiplication (that is, they satisfy Axiom M-l).

X +2 and x and x +2<0 +2<0 < - 2. and x 5 Since there are no numbers x satisfying both conditions x> 11/5 and x < - 2, the set S2 is empty. 7) consists of Sl = ( - 2, Il/5). 7 We shall assume the reader is familiar with the notion of absolute value and elementary manipulations with equations and inequalities involving the absolute value of numbers. 1. 2. PROBLEMS 1. Consider the field consisting of all numbers of the form a + bV7 where a and b are rational. Does this field satisfy Axiom I? Justify your answer.

J(x) = {( ~~: ~ ) 1/2 x2 - 9 12. I(x) 3

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A First Course in Real Analysis by M. H. Protter C. B. Morrey Jr.


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