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Analysis1991Aug - GRADUATE PRELIMINARY EXAMINATION ANALYSIS...

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Unformatted text preview: GRADUATE PRELIMINARY EXAMINATION ANALYSIS Fall 1991 Instructions: Do all problems. Each problem is worth 10 points. 1. Show that every uncountable subset of the real numbers has a limit point. ‘2. The sequence of real numbers {2"} is defined recursively by :1 = 1 and 1 3 zn+1 = (In+$%)/ . Prove that x" converges, and find the limit. 3. Let {fn} be a sequence of continuous functions defined on a compact metric space K, and suppose f1, converges uniformly on K to a function f Prove that f3, converges uniformly to f2 on K. 4. Prove the following: if f is a continuous, real valued function on [0,1] such that f (0) = 0 and 1 [znflzflx :0 for n=1,2,..., o 1 then m) = o for all :r 6 [0,1]. Hint: Show that ] f2(z)d:r = o. 0 5. Let F(z, y, z) = 31: + 2y + z — ysin(zz). (a) Can the equation F(:z:,y, z) = 0 be solved for z = f(z,y) in a neighborhood of the point (0,—1) satisfying f (0, —l) = 2? Justify your answer. (b) State a precise version of what is asked for in (a). [Be as complete as possible. 6. The function f maps [0,1] onto [0,1], and is monotone. Prove f is continuous on [0,1]. ...
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