Solved by Expert Tutors
Solved by Expert Tutors
Question

# About Integral Calculus's question. The text book I used is &amp;quot;Advanced

calculus&quot; by Folland. Chapter 1 maybe helpful. The questions, PDF of the &quot;Advanced calculus&quot; '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 Instructor’s Solution Manual for ADVANCED CALCULUS Gerald B. Folland  Show entire document

m ipsum dolor sit amet, consectetur adipiscing elit. Nam lacinia pulvinar tortor nec facilisis. Pellentesque dapibus efficitur laoreet. Nam risus ante, dapibus a molestie consequat, ultr
facilisis. Pellentesque dapibus efficitur laoreet.

e vel laoreet ac, dictum vitae odio. Donec aliquet. Lorem ipsum dolor sit amet, consec , dictum vitae odio. Donec al ctum vitae odio. Donec aliquet. Lorem ipsum dolor sit acinia pulvinar tortor nec facilisi dictum vitae odio. Donec aliquet. Lorem ipsum dolor sit amet, consectetur adip

#### Subscribe to view the full answer

Subject: Calculus, Math

### Why Join Course Hero?

Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors.

• ### -

Study Documents

Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.

Browse Documents
• ### - 