Theorem 7. Every element of R has an additive inverse. More specificially, if x = [(a,)], then -æ = [(-a;)]. Proof. Let æ E R. Write a as [(a;)] where (a;) is Cauchy in Q. By Lemma 6 the sequence (-a;) is also Cauchy, so y = [(-a;)] is a real number. We leave it to the reader to show that y is the additive inverse of x. SO
Theorem 7. Every element of R has an additive inverse. More specificially, if x = [(a,)], then -æ = [(-a;)]. Proof. Let æ E R. Write a as [(a;)] where (a;) is Cauchy in Q. By Lemma 6 the sequence (-a;) is also Cauchy, so y = [(-a;)] is a real number. We leave it to the reader to show that y is the additive inverse of x. SO
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...
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