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Unformatted text preview: space is [1, 2] (with the usual Euclidean metric). YOU NEED NOT PROVE THAT ℝ IS COMPLETE. 3. Let X be a metric space with metric d(x,y). a) State the CMP. (Don’t forget any hypotheses or conclusions!) b) Prove the uniqueness part of the CMP. c) Prove convergence of the sequence. d) Prove the limit point is a fixed point. 4. a) Suppose S is a finite set with a metric d and suppose A is a nonempty subset of S. Prove A is closed. b). Let F(x) = xsin(1/x) when x ≠ 0 and let F(0) = 0. Prove or disprove that F is uniformly continuous on [-1,1]. c) Show there is a proper subset of ℚ with ∂A = ∅ . (Here ℚ is the metric space, with the usual Euclidean metric.) 5. a) Find the radius of convergence of the power series ∑ (n!) 3 (2n!)-1 z 2n (Notice the power of z is 2n.) b) Suppose the power series ∑ a n z n has radius of convergence R > 0. Prove that if 0 < r < R, there is a real number K such that |z| ≤ r ⇒ ∑ |a n z n | ≤ K....
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- Math, Metric space, CMP