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Real Analysis Qualifying Exam Date: September 21, 2014 Duration: 3 Hours Instructions: This exam consists of 5 questions. Each question is worth 5 points giving a grand total of 25 points possible. Please present all of your work in a clear and concise manner and answer each question as completely as possible. Unsupported work will receive no credit and partially completed work may receive partial credit. Good luck! 1. Let K, F ⊆ R be nonempty subsets of real numbers. Define the distance between K and F by d(K, F ) := inf{|x − y| : x ∈ K, y ∈ F }. (a) Suppose K is compact and F is closed. Show that if d(K, F ) = 0, then K ∩ F 6= ∅. (b) Give an example (with justification) of 2 nonempty, closed subsets K, F ⊆ R such that d(K, F ) = 0 but K ∩ F = ∅. 2. Let {an } ⊂ R be a bounded sequence of real numbers and suppose {bn } is a sequence of real numbers such that bn+1 < bn for all n ∈ N and lim bn = 0. Show that the series n→∞ ∞ X an (bn − bn+1 ) n=0 converges. 3. (a) Let f : R → R be a uniformly continuous function. Show that if {xn } ⊂ R is a Cauchy sequence of real numbers, then {f (xn )} is a Cauchy sequence. (b) Suppose that {fn } is a sequence of continuous functions that converge uniformly on a subset A ⊂ R to a function f . Show that f is continuous on A. 4. Let 0 < < 1. For each n ∈ N∪{0}, consider the function fn : R → R defined by fn (x) = (a) Determine the pointwise limit of the series of functions ∞ X x2 . (1 + sin2 x)n fn (x) for x ∈ [, 1]. n=0 (b) Show that the series of functions ∞ X fn (x) converges uniformly on [, 1]. n=0 (c) Show that the series of functions ∞ X fn (x) does NOT converge uniformly on [0, 1]. n=0 5. (a) State the definition for a real-valued function f : [a, b] → R to be Riemann integrable on the interval [a, b]. (b) Let f, g : [a, b] → R be Riemann integrable functions and suppose that the set E = {x ∈ (a, b) : f (x) 6= g(x)} Z b Z is finite. Use the definition of Riemann integrability to show that f (x) dx = Hint: Consider the function f − g. a b g(x) dx. a Note: If you choose to work with a definition of Riemann integrability different than that stated in part (a), please provide this alternate definition.