solutionshw9-201141

solutionshw9-201141 - Solutions Homework 9. (1) Let cn >...

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Solutions Homework 9. (1) Let c n > 0 in R . Prove that lim c n +1 c n lim n c n lim n c n lim c n +1 c n . Solution: Let A = lim c n +1 c n and let ± > 0. If A = , then there is nothing to prove, so assume A < . Then there exists N such that c n +1 c n < A + ± for all n N . This implies that c N + m < ( A + ± ) c N + m - 1 < ··· < ( A + ± ) m c N for all m 1. From this we conclude that for n = N + m we have n c n < ( A + ± ) ± c N ( A + ± ) N ² 1 n . This implies lim n c n A + ± for all ± > 0 and it follows that lim n c n lim c n +1 c n . The left hand side of the inequality can be proved similarly, or can be deduced from the right hand side, if we apply the right hand side inequality to the sequence d n = 1 c n and use that lim n d n = 1 lim n c n . (2) Let a n 0 and b n 0. Assume that both ( a n ) and ( b n ) are bounded sequences. (a) Prove that
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solutionshw9-201141 - Solutions Homework 9. (1) Let cn &gt;...

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