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Math 414: Analysis I Homework 4 Due: February 19th, 2014 Name: The following problems are for additional practice and are not to be turned in: (All problems come from Basic Analysis, Lebl ) Exercises: 2.1.1-2.1.5, 2.1.9-2.1.11, 2.1.13, 2.1.14, 2.1.17, 2.1.18 Turn in the following problems. 1. Show that if (xn ) is unbounded, then there exists a subsequence (xnk ) such that lim(1/xnk ) = 0. √ 2. Let x1 := 1 and xn+1 := 2 + xn for n ∈ N. Show that (xn ) is convergent and find the limit. Hint: Consider the Monotone Convergence Theorem here. 3. Suppose that (an ) converges to A, and define a new sequence (bn ) by bn = for all n ∈ N. Prove that (bn ) converges to A. an + an+1 2 4. Suppose that (xn ) converges to x0 and (yn ) converges to x0 . Define a sequence (zn ) as follows: z2n = xn and z2n−1 = yn . Prove that (zn ) converges to x0 . 5. Exercise 2.1.6 from Basic Analysis, Lebl 6. Exercise 2.1.7 from Basic Analysis, Lebl 7. Exercise 2.1.12 from Basic Analysis, Lebl 8. Exercise 2.1.15 from Basic Analysis, Lebl 1