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hw1 - Y α P = u α ∈A f-1 Y α f-1 p i α ∈A Y α P =...

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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. Define limsup n →∞ E n = { x : x E n for infinitely many n } , liminf n →∞ E n = { x : x E n for all but finitely many n } . Prove that limsup n →∞ E n = intersectiondisplay k =1 uniondisplay n = k E n , liminf n →∞ E n = uniondisplay k =1 intersectiondisplay 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 parenleftBigg uniondisplay α ∈A
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Unformatted text preview: Y α P = u α ∈A f-1 ( Y α ) , f-1 p i α ∈A Y α P = i α ∈A f-1 ( Y α ) . 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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