By Joseph R. Lee

ISBN-10: 0124407501

ISBN-13: 9780124407503

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Extra info for Advanced calculus with linear analysis

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Where „ fc^ + l a*(n) = k: n 1 , 1 = k7 + 7-· kn Find x = limn^oo xn. Show x n G (co) and x € (c 0 ). 8. Let xn = {a Ä (n) }, where ak(n) = nk + k + n nk for n = 1, 2, . . , fc = 1, 2, . . Find x = lining x n . Show o:n G (c) and x Ç (c) but xn Í (c0) and x Í (co). 9. The elements {e¿} in the space (c0) form a 6

3 THEOREM Every bounded monotone sequence of numbers converges. real Proof We prove this for the case that {an} is nondecreasing and leave the others as problems. Since {an\ is bounded above, there exists a supremum, call it a. Then, an < a for every n. Also, for e > 0, there is at least one number in the f Landau, E . "Foundations of Analysis" 2nd ed. Chelsea, Bronx, New York, 1960. X Such a system is referred to as a complete ordered field. III. 44 Completeness Properties sequence, call it αχ, such that a^ > a — e.

7. *«»}, X2 = {a*<2>}, . . , x» = {a*(w)} for Λ = 1, 2, . . , where „ fc^ + l a*(n) = k: n 1 , 1 = k7 + 7-· kn Find x = limn^oo xn. Show x n G (co) and x € (c 0 ). 8. Let xn = {a Ä (n) }, where ak(n) = nk + k + n nk for n = 1, 2, . . , fc = 1, 2, . . Find x = lining x n . Show o:n G (c) and x Ç (c) but xn Í (c0) and x Í (co). 9. The elements {e¿} in the space (c0) form a 6