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Unformatted text preview: Math508, Fall 2010 Jerry L. Kazdan Problem Set 3 DUE: Thurs. Sept. 30, 2010. Late papers will be accepted until 1:00 PM Friday . 1. Find all (complex) roots z = x + iy of z 2 = i . 2. Let x n > 0 be a sequence of real numbers with the property that they converge to a real number c > 0. Prove there is a real number m > 0 such that x n > m for all n = 1 , 2 ,... . 3. Let x k ∈ R , x k = 0 be a sequence of real numbers. If x k → c = 0, show that 1 / x k → 1 / c . 4. Calculate lim n → ∞ √ n 2 + n n . 5. If c > 0, show that c n n ! → 0 as n → ∞ . 6. Let a n be an increasing sequence of real numbers that is bounded above, so there is an M such that a n < M for all n = 1 , 2 , 3 ,... . Show there is a real number A such that a n → A . 7. Let p k = ( x k , y k ) ∈ R 2 , k = 1 , 2 ,... be a sequence of points in the plane (with the usual Eu clidean metric). Show that { p k } converges to p = ( x , y ) if and only if x k → x and y k → y ....
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
 Fall '10
 STAFF
 Real Numbers

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