real_analysisii_homework5-1 - Real Analysis II Homework 5...

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Real Analysis II Homework 5 ector Guillermo Cu´ ellar R´ ıos March 9, 2006 18.2 Mark each statement True or False. Justify each answer. (a) If a convergent sequence is bounded, then it is monotone. False. The convergent sequence sin( n ) n 2 is bounded and it is not mono- tone. (b) If ( s n ) is an unbounded increasing sequence, then lim s n = + True by theorem 18.8. 18.3 Prove that each sequence is monotone and bounded. Then find the limit. (b) s 1 = 2 and s n +1 = 1 4 ( s n + 5) for n N We shall show that ( s n ) is a bounded decreasing sequence. Computing the first three terms of the sequence, we find s 1 = 2 s 2 = 1 4 (2 + 5) = 1 . 75 s 3 = 1 4 7 4 + 5 1 . 69 . It appears that the sequence is bounded below by 1. To see if this conjecture is true, let us try to prove it using induction. Certainly, s 1 = 2 > 1. Now suppose that s k > 1 for some k N . Then s k +1 = 1 4 ( s k + 5) > 1 4 (1 + 4) = 17 4 > 1 . Thus we may conclude by induction that s n > 1 for all n N . To verify that ( s n ) is a decreasing sequence, we also argue by induction. Since s 1 = 2 and s 2 = 1 . 75, we have s 1 > s 2 , which establishes the 1
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basis for induction. Now suppose that s k > s k +1 for some k N . Then we have s k +1 = 1 4 ( s k + 5) > 1 4 ( s k +1 + 5) = s k +2 . Thus the induction step holds and we conclude that s n > s n +1 for all n N . Thus ( s n ) is a decreasing sequence and it is bounded by the interval [1,2]. We conclude from the monotone convergence theorem (18.3) that ( s n ) is convergent. Since lim s n +1 = lim s n , we see that s must satisfy the equation s = 1 4 ( s + 5)
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