Calculus

Download PDF by Gert K. Pedersen: Analysis Now

By Gert K. Pedersen

ISBN-10: 0387967885

ISBN-13: 9780387967882

ISBN-10: 3540967885

ISBN-13: 9783540967880

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Example text

16. TRICHOTOMY LAW. For arbitrary real numbers a and b, exact@ one of the three relations a < b, b < a, a = b holds. Zf a < b andb < c, then a < c. 17. 18. If a < b, then a + c < b + c. 19. 20. THEOREM If If LAW. a < b and c > 0, then ac < bc. a # 0, then a2 > 0. 21. 1 > 0. 22. Zf a < b and c < 0, then ac > bc. 23. If a < b, then -a > -b. Znparticular, fa < 0, then -a > 0. 24. If ab > 0, then both a and b are positive or both are negative. 25. If a < c and b < d, then a + b < c + d. Again, we shall prove only a few of these theorems as samples to indicate how the proofs may be carried out.

From this, we concluded that the assertion is true for a11 positive integers. The idea of induction may be illustrated in many nonmathematical ways. For example, imagine a row of toy soldiers, numbered consecutively, and suppose they are SO arranged that if any one of them falls, say the one labeled k, it Will knock over the next one, labeled k + 1. Then anyone cari visualize what would happen if soldier number 1 were toppled backward. It is also clear that if a later soldier were knocked over first, say the one labeled n, , then a11 soldiers behind him would fall.

A/b) + (C/d) = (ad + bc)/(bd) zf b # 0 and d # 0. 4. Those readers who are interested may find it instructive to carry out proofs of the remaining theorems. 1. Given a + b = a + c. By Axiom 5, there is a numbery such that y + a = 0. Since sums are uniquely determined, we have y + (a + 6) = y + (a + c). Using the associative law, we obtain (y + a) + b = (y + a) + c or 0 + b = 0 + c. But by Axiom 4 we have 0 + b = b and 0 + c = c, SO that b = c. Notice that this theorem shows that there is only one real number having the property of 0 in Axiom 4.

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Analysis Now by Gert K. Pedersen


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