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Homework #11 Subsequences Read section 2.6 Exercises from the text: Page 110: #1, #2, #4, #7, Other exercises in thought and logic that you will most certainly find interesting. 1. Suppose that an converges to A and that bn converges to A. Define the sequence cn by that 2. c2n an and c2n1 bn . Prove that cn converges to A. Let { f n } be the Fibonacci sequence f n : 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , ... satisfying f1 = f2 = 1 and fn+1= fn+ fn-1. Let {rn } be the sequence of ratios of successive Fibonacci numbers. So rn f n 1 and fn rn : 1 , 2 , 3/2 , 5/3 , 8/5 , 13/8 , 21/13 , ... a. Is this sequence monotonic? b. Prove that rn+1 = 1 + 1/rn and hence prove that if this sequence were convergent then its limit would be ( 5 + 1)/2. c. Prove that rn+2 = (2rn + 1)/(rn + 1). Hence show that the subsequence of odd terms and the subsequence of even terms are monotonic and bounded. d. Deduce that (rn) is convergent to the Golden Ratio. 3. Show that every sequence has a monotone subsequence. 4. Let an be any sequence and suppose that S lim sup an . Prove that there exists a subsequence of an converging to S. n 5. Let a be a real number in (0, 1). Let an be the real number obtained by deleting the first n digits of the decimal expansion of a. For example, if a = 0.1234567... then a1 = 0.234567.., a2 = 0.34567..., etc. For which a (0,1) is the sequence an convergent and what is its limit ?