Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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Unformatted text preview: 13.12 This is false. Let S = { , 1 } and set x n = 1 ,y n = 0. 13.23 Since sup A a and sup B b for every a A and b B , sup A +sup B a + b for every a + b C . Thus sup A +sup B is an upper bound for C . By Proposition 13.15, there are sequences h a n i in A and h b n i in B such that a n sup A and b n sup B . Then h a n + b n i is a sequence in C with a n + b n sup A + sup B . By Proposition 13.15, sup A + sup B = sup C . 13.29 First we prove that h x n i converges using the Monotone Convergence Theorem. Clearly 0 x n . On the the other hand, x n = 1 + n 1 + 2 n = 1 n + 1 1 n + 2 1 n + 1 2 1 + 1 2 = 2 . Therefore h x n i is bounded. Further, for every n N we have x n +1- x n = 1 + ( n + 1) 1 + 2( n + 1)- 1 + n 1 + 2 n = 2 + n 3 + 2 n- 1 + n 1 + 2 n = (2 + n )(1 + 2 n )- (1 + n )(3 + 2 n ) (3 + 2 n )(1 + 2 n ) = (2 n 2 + 5 n + 2)- (2 n 2 + 5 n + 3) (3 + 2 n )(1 + 2 n ) =- 1 (3 + 2 n )(1 + 2 n ) < , so h x n i is monotone. By the Monotone Convergence Theorem,is monotone....
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HW16solns - 13.12 This is false. Let S = { , 1 } and set x...

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