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# Note for MATH 425A with Professor Dawson at UA

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COURSE
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TYPE
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KARMA
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
1 For each natural number n and each number x 6 01 de ne fnx z 7 x a Find the function f 01 a R to which the sequence fn 01 a R converges pointwise Hint Look at z 01 1 and z 6 01 b Prove or disprove that the convergence is uniform Hint Prove by contradiction If the convergence is uniform what do we know about the limit 2 Suppose Zak converges and that 1k 2 0 for all k E N Let p gt 1 Prove that 20 k0 k0 COHVngSS Hint Since we know the series converges what do we know about the sequence ak How can you use this to show 11017 lt 1k for k suf ciently large Now prove the 00 partial sums of 20 are bounded and monotone k0 3 For each natural number n and each number x E R de ne fnx 1 sinnx a Show that fn converges uniformly to the function f1 0 V1 6 R Hint Use that sinx S 1 for all x b Notice that each fn is differentiable everywhere Do not prove this Show that the pointwise limit of need not exist and explain why this does not contradict the theorem regarding uniformly convergent sequences of differentiable functions Theorem 933 Hint Find the derivative of Then x a value of z E 0 27139 and look at the values of You should be able to choose 1 so that it is easy to see that doesn7t converge 4 Let f and g be continuous real valued functions on 11 such that fr gr for each 7 E Q 11 Prove that f1 91 for all z 6 11 Hint De ne 111 fx 7 Then 111 is continuous Prove the desired result by contradiction 5 Prove that sinx isiny S 17y for all Ly E R Hint use Mean Value Theorem Then use this and the de nition of uniform continuity to prove that sinz is uniformly continuous on R Hint Use that cos1 S 1 for all z and the Mean Value Theorem Then use the epsilon delta de nition of uniform continuity 00 00 6 Prove that if 2 an converges and bn is a bounded sequence then Zanbn con n0 n0 verges 00 00 Hint Let 5 and tn be the partial sums of Z lanl and Zanbn respectively Prove n0 n0 each is Cauchy Let an be a Cauchy sequence Suppose can has a convergent subsequence am converging to a number a Prove using the de nition of convergence that an a a Hint This is a typical 62 proof Write down the de nition of Cauchy and convergence using epsilons Suppose that the function f R a R has the property that 7x2 S f 3 2 Vs E R Prove that f is differentiable at z 0 and that f 0 0 Hint Write down the limit you are trying to show to prove that the derivative at 0 exists Then use the inequality given to bound Suppose that the function f 17 a R is de ned by f c for all z 6 ab and fa d E R Use the de nition of the integral7 the Archimedes Riemann Theorem7 or the Cauchy Criterion for integrals to show that f is integrable on 17 b and fcbia Hint For the Archimedes Riemann theorem7 de ne a sequence of partitions7 Pn of 17 with 3 partition points ammb Choose xn so that a a Look at UfPn and Lf7 P for each n

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