s4 - Math 8601: REAL ANALYSIS. Fall 2010 Homework #4....

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Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #4. Problems and Solutions. #1. Let R n be represented in the form R n = S k =1 I k , where { I k } are non-overlapping cubes with edge length 1. Let F k be a closed subset of I k , k = 1 , 2 ,... . Show that the set F = S k =1 F k is closed. Proof. Let x be a limit point of F = S k =1 F k , i.e. x = lim x j , where { x j } F . We know that any convergent sequence is bounded, therefore { x j } F N [ k =1 I k = N [ k =1 ( F I k ) = N [ k =1 F k =: F for some N . Since F is closed, x = lim x j F F , and F is closed. #2. Give an example of families of sets { A } , { B } , where I- an arbitrary set, for which \ A \ B " \ A B . Solution. Let A be an arbitrary nonempty set. Take A 1 = A 2 = B 1 = A, B 2 = . Then 2 \ =1 A 2 \ =1 B = A = A 6 = , 2 \ =1 A B = ( A A ) ( A ) = A = ....
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This note was uploaded on 02/15/2012 for the course MATH 8601 taught by Professor Staff during the Fall '08 term at Minnesota.

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s4 - Math 8601: REAL ANALYSIS. Fall 2010 Homework #4....

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