1873_solutions

If x is any partial limit of z n then the number x1 x

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Unformatted text preview: ven about A and B we know that sup A inf B. In the event that sup A  inf B we know that for every sequence x n in A and every sequence y n in B we have y n x n inf B sup A  0 for every n and so we can’t have y n x n 0 as n Ý. Therefore if condition a is false then so is condition b. Suppose now that condition a is true; in other words, that sup A  inf B. For every positive integer n, we use the fact that sup A 1 n is not an upper bound of A to choose a member x n of A such that sup A 1  x n . n For every positive integer n we use the fact that inf B  1 n is not a lower bound of B to choose a member y n of B such that y n  inf B  1 . n Thus for each n we have sup A 1  x n y n  inf B  1 n n and therefore 0 yn xn  2 n from which it follows that y n x n 0 as n Ý. 8. Suppose that S is a nonempty bounded set of real numbers. Prove that there exist two sequences x n and y n in the set S such that y n x n sup S inf S as n Ý. We have already seen that there exists a sequence x n in S such that x n sup S as n Ý. In the same way we can see that there exists a sequence y n in S such that y n inf S as n Ý. We now have y n x n sup S inf S 144 as n Ý. Exercises on Monotone Sequences 1. Given that c  1, use the following method to prove that c n Ý: a. Write   c 1, so that c  1  , and then use mathematical induction to prove that, if n is any positive integer then c n 1  n. For each positive integer n we take p n to be the assertion that c n 1  n. Since c  1   we know that the assertion p 1 is true. Now suppose that n is any positive integer for which the assertion p n is true. We observe that 1   1  n c n1  cc n  1  n  1   n 2  1  n  1  and so the assertion p n1 is also true. We deduce from mathematical induction that the inequality c n 1  n is true for every positive integer n. b. Explain why 1  n Ý and then use this exercise in this subsection to show that c n Ý. To prove that 1  n Ý as n Ý, suppose that w is a real number. The inequality 1  n  w will hold when n  w 1.  We choose a positive integer N such that N w 1  and we observe that 1  n  w whenever n N. Ý: 2. Given that c  1, use the following method to prove that c n cn is increasing and deduce that it has a limit. a. Explain why the sequence The fact that the sequence c n is increasing follows at once from the inequality c n1  cc n  1c n . It now follows from the monotone sequences theorem that the sequence c n has a limit. b. Call the limit x and show that if x is finite then the equation c n1  cc n leads to the equation x  cx, which implies that x  0. But x cannot be equal to zero? Why not? From the equation c n1  cc n we obtain lim lim c n1  n Ý cc n nÝ which gives us x  cx. Since c n  1 for every n we know that x 3. Suppose that |c |  1 and that, for every positive integer n, n xn  ci 1. i1 Explain why xn From the identity 145 1 1 c . 1 and therefore x 0. n 1 ci c 1 1 cn i1 we deduce that n ci i1 1 n 1 c 1c...
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