hw4 - F ⊂ R find a sequence of reals x n whose set of...

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Modern Analysis, Homework 4 Due July 1, 2010 1. [12] (5 points) Prove that every sequence of reals has a monotone (ie either decreasing or increasing, though not necessarily strictly ) subsequence. 2. [15] (10 points) Rudin, Ch 3, problem 16, parts (a) and (b). 3. [25] (10 points) Rudin, Ch 3, problem 14, parts (a)–(d) (NOT (e)). 4. [12] (5 points) Rudin, Ch 3, problem 21. 5. [12] (5 points) Rudin, Ch 3, problem 22. The following problems are optional : 6. [30] (5 points) Given an arbitrary closed subset
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Unformatted text preview: F ⊂ R , find a sequence of reals { x n } whose set of subsequential limits is precisely F . Hint: Using homework 3, show that F is separable, when given the induced metric by R . Rudin, Ch3, problem 25 might be helpful. 7. [32] (5 points) Show that the Cantor set consists precisely of those points in [0 , 1] having a base-3 expansion with no 1. Notice this gives an alternate proof that the Cantor set is uncountable! 1...
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