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About Integral Calculus's question. The text book I used is "Advanced

calculus" by Folland. Chapter 1 maybe helpful. The questions, PDF of the "Advanced calculus" 's solution manual are in the attachments. After I accpeted the solutions, I will give the amount $65 as tips.

MATH 2000A - Fall 2014 Assignment 1 Due beginning of class on Friday, September 19. 1. Determine if the following sets are open, closed or neither and justify your answer. (a) R \ N (b) { x R 2 | x 1 + x 2 = 1 } ⊂ R 2 (c) ( a 1 ,b 1 ) × ··· ( a n ,b n ) = { x R n | a k < x k < b k for k = 1 ,...,n } ⊂ R n (d) { x R n | x 1 + ··· + x n Q } ⊂ R n 2. (a) Let S R n with S 6 = and S 6 = R n . Prove that ∂S 6 = . (b) Prove that if S R n is simultaneously open and closed, then either S = or S = R n . 3. Prove that a subset of R n is open if and only if it can be written as a union of open balls. 4. Let A be an index set (e.g. A = { 1 ,...,n } or A = N or A = R ). For each α A , let K α R n be a closed set. (a) Prove that α A K α is closed, i.e. the intersection of closed sets is closed. (b) Give an example to show that α A K α is not necessarily closed, i.e. the union of closed sets is not necessarily closed. (c) Prove that ( α A K α ) c = α A K c α . Explain why this implies that the union of open sets is open. 5. Let ( x n ) and ( y n ) be Cauchy sequences in R n , i.e. lim n,m | x n - x m | = 0 and lim n,m | y n - y m | = 0. For each n , let d n = | x n - y n | . Prove that d n is a Cauchy sequence in R . 6. Let f : R n R m be a continuous function and define g : R n R by g ( x ) = | f ( x ) | , x R n . Prove that g is also continuous. 1
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Instructor’s Solution Manual for ADVANCED CALCULUS Gerald B. Folland
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calculus.pdf
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Subject: Calculus, Math

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