1873_solutions

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 n1  cc n  1  n  1   n 2  1  n  1  and so the assertion p n1 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 n1  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 n1  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 n1  cc n we obtain lim lim c n1  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. i1 Explain why xn From the identity 145 1 1 c . 1 and therefore x 0. n 1 ci c 1 1 cn i1 we deduce that n ci i1 1 n 1 c 1c...
View Full Document

Ask a homework question - tutors are online