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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 nonoverlapping 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 α } , { B α } , where α ∈ I an arbitrary set, for which \ α A α Δ \ α B α " \ α A α Δ B α . Solution. Let A be an arbitrary nonempty set. Take A 1 = A 2 = B 1 = A, B 2 = ∅ . Then 2 \ α =1 A α Δ 2 \ α =1 B α = A Δ ∅ = A 6 = ∅ , 2 \ α =1 A α Δ B α = ( A Δ A ) ∩ ( A Δ ∅ ) = ∅ ∩ A = ∅ ....
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 Fall '08
 Staff
 Math, Continuous function, Metric space, measure, FN, Lebesgue measure

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