IA08WS5 - Introduction to Analysis: Fall 2008 Practice...

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Introduction to Analysis: Fall 2008 Practice problems V MTH 4101/5101 10/21/2008 1. Show that the sequence { 1 ( n 2 +1) } converges to 0. Solution: Let ± > 0 be given. For n IN , we have 1 n 2 + 1 < 1 n 2 1 n . Choose N such that 1 N < ± . Then we have, | 1 n 2 +1 - 0 | < 1 n < ±. 2. Show that the sequence { 3 n +2 ( n +1) } converges to 3. Solution: Let ± > 0 be given. We ± ± ± ± 3 n + 2 n + 1 - 3 ± ± ± ± = ± ± ± ± 3 n + 2 - 3 n - 3 n + 1 ± ± ± ± = ± ± ± ± - 1 n + 1 ± ± ± ± = 1 n + 1 < 1 n . 3. If 0 < b < 1 show that lim n →∞ b n = 0. Solution; If ± > 0 is given, we have b n < ± n ln b < ln ± n > ln ± ln b . (The last inequality is reversed because ln b < 0.) 4. Let { x n } be a sequence of real numbers and let x IR. If { a n } is a sequence of positive real numbers, with lim n →∞ a n = 0 and if for some constant C > 0 and some m IN we have | x n - x | ≤ Ca n for all n m show that lim x n = x . Solution: Let ±. 0 be given. Since lim a n = 0 we know that there exists K such that such that n K implies a n = | a n - 0 | < ± C
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IA08WS5 - Introduction to Analysis: Fall 2008 Practice...

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