hw1 - Y P = u A f-1 ( Y ) , f-1 p i A Y P = i A f-1 ( Y ) ....

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Math 240A: Real Analysis, Fall 2011 Homework Assignment 1 Due Friday, September 30, 2011 1. Let { E n } n =1 be a sequence of sets. DeFne lim sup n →∞ E n = { x : x E n for inFnitely many n } , lim inf n →∞ E n = { x : x E n for all but Fnitely many n } . Prove that lim sup n →∞ E n = i k =1 u n = k E n , lim inf n →∞ E n = u k =1 i n = k E n . 2. Let X and Y be two sets and f : X -→ Y a mapping. Let { Y α } α ∈A be a family of subsets of Y . Prove f - 1 p u α ∈A
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Unformatted text preview: Y P = u A f-1 ( Y ) , f-1 p i A Y P = i A f-1 ( Y ) . 3. ind a bijection from N to N 2 . 4. Construct a sequence of open sets U n ( n = 1 , 2 , . . . ) of R such that n =1 U n is not open. 5. Prove the statements ii, iii, and iv on Page 13 of the textbook....
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This note was uploaded on 12/11/2011 for the course MATH 240a taught by Professor Rothschild,l during the Fall '08 term at UCSD.

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