By Abramowitz M., Stegun I.A. (eds.)
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By the same token, you should find as you study this book that your understand ing of previously learned mathematics is broadened and deepened. In mathematics, truly, nothing should be forgotten. 4 The Rational Number System We take the rational number system as our starting point in the con struction of the real number system. We could, of course, give a detailed construction of the rational numbers in terms of more primitive notions. 4 The Rational Number System 19 common notions that are accepted without formal development.
Then by taking m to be the larger of m l and m2 , we have Proof: . . for j, k > m. For part b, given any error lin, there exists ml such that IXk-X� 1 < 1/2n for k > m l and there exists m2 such that I Yk - Y�I < 1/2n for k > m 2 , because of the equivalence of Xl , X 2 , . . and xl ' x2" " and the equivalence of Y I , Y2 , and � , y�, . . If we take m to be the larger of m l and m2 , then we have . . for k > m . QED The real number X + Y is the equivalence class of the Cauchy sequence X l + Y I , X 2 + Y2 , .
In addition, let . • • . . . The argument is very similar to the proof of the transitivity of equivalence given in the last section. For part a, given any error lin, there exists m l such that IXj - xk l < 1/2n for j, k > ml and there exists m 2 such that IYi - Yk I < 1/2n for j, k > m2 , because Xl , X 2 , . and YI , Y2 , . are Cauchy sequences. Then by taking m to be the larger of m l and m2 , we have Proof: . . for j, k > m. For part b, given any error lin, there exists ml such that IXk-X� 1 < 1/2n for k > m l and there exists m2 such that I Yk - Y�I < 1/2n for k > m 2 , because of the equivalence of Xl , X 2 , .
Handbook of mathematical functions (without numerical tables) by Abramowitz M., Stegun I.A. (eds.)