hw5 - MATH 444 ELEMENTARY REAL ANALYSIS HOMEWORK 5 Due date...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 5 Due date: Oct 1 (Wed) Exercises from the textbook. 3.2: 15; 19(b)(d); 20; 22 3.3: 3; 6 Out-of-textbook exercises. 1. Let ( y n ) be a sequence of nonzero reals and suppose lim n →∞ y n = y 6 = 0. Follow the steps (i)–(iii) below to prove that lim n →∞ 1 y n = 1 y . (i) Show that l := inf {| y n | : n N } > 0. (ii) Deduce from (i) that the set { 1 | y n | : n N } is bounded above. What is the supremum of this set? (iii) Prove that lim n →∞ 1 y n = 1 y . 2. Let ( x n ) be a sequence of nonzero reals such that lim n →∞
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Unformatted text preview: +1 x n ± ± ± ± = L > 1 . Follow the steps (i)–(iii) below to prove that ( x n ) is unbounded, and hence divergent. (i) Prove that for any r < L , there is N r ∈ N such that ∀ n ≥ N , | x n +1 | > r | x n | . (ii) Fix a real r ∈ (1 ,L ) and let N r be as in part (a). Using mathematical induction prove that for all n ≥ N r , | x n | ≥ r n-N r | x N r | . (iii) Conclude that ( x n ) is unbounded, and hence divergent....
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  • Spring '08
  • JUNGE
  • Math, lim, elementary real analysis, nonzero reals, lim yn

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