By M. H. Protter C. B. Morrey Jr.
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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. 2. Consider the set of all numbers of the form aiV7 where a is a real number and i = V-=T . Show that it is possible to give this set an ordering in such a way that it satisfies Axiom I. Does this set form a field? 3. Show that the set of all complex numbers a + bi, with a and b rational, satisfies all the axioms for a field. In Problems 4 through 7 find in each case the solution set as an interval, and plot.
What conclusion can lim j[g(x)] x~a+ when a = TT/4? Same problem when a = TT/2, when a = O. 14. 4 on the limit of equal functions for limits from the left. 15. 5 on the limit of a sum for limits from the right. 16. 6 on the limit of a product for limits from the right. 17. 9 on the limit of inequalities for limits from the left. 18. 10 (the Sandwiching theorem) for limits from the right. 5. , x tends to infinity to the right) or when x goes to infinity through decreasing negative values (x tends to infinity to the left).
Prove by induction that (XI k + X2 + ... + 2(XIX2 + XIX) + ... + XIXk 29 I: The real number system 16. 24. 17. 25. 18. 26. 19. 27. ] 20. 28. ] 21. 29. 22. 2. 23. 2. 24. 2. 5 We denote by N x N the set of all ordered pairs of natural numbers (m, n). State and prove a Principle of mathematical induction for sets contained in NXN. 5 An asterisk is used to indicate difficult problems. 1 Continuity Most of the functions we study in elementary calculus are described by simple formulas. These functions almost always possess derivatives and, in fact, a portion of any first course in calculus is devoted to the development of routine methods for computing derivatives.
A First Course in Real Analysis by M. H. Protter C. B. Morrey Jr.