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

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