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Exercise 2 Let (An)n be a countable collection of subsets of R such that m*(An) <∞. Define the set (1) E = {r ER: * An for infinitely many n}. Show that E = {xER: VnEN, 3no>n: x € Ang}. (2) Let En Un Ak = An U An+1 U.... Show that ECE, VnE N. = (3) (a) Show that (b) Deduce that n=1 m²(E) ≤ [m²(A), VEN. k=n m* (E) = 0. (Hint: If a series converges, the remainder tends to 0)

Exercise 2 Let (An)n be a countable collection of subsets of R such that m*(An) <∞. Define the set (1) E = {r ER: * An for infinitely many n}. Show that E = {xER: VnEN, 3no>n: x € Ang}. (2) Let En Un Ak = An U An+1 U.... Show that ECE, VnE N. = (3) (a) Show that (b) Deduce that n=1 m²(E) ≤ [m²(A), VEN. k=n m* (E) = 0. (Hint: If a series converges, the remainder tends to 0)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 2 Let (An)n be a countable collection of subsets of R such that m*(An) < 0. Define the set (1) Show that E = {xER: € An for infinitely many n}. E = {x ER: VnEN, 3no >n: x € Ang}. (2) Let En Un Ak = An U An+1 U.... Show that = EcEm VneN. (3) (a) Show that (b) Deduce that 80 7=1 m²(E) ≤ [m*(A), VEN. m* (E) = 0. (Hint: If a series converges, the remainder tends to 0)
Transcribed Image Text:Exercise 2 Let (An)n be a countable collection of subsets of R such that m*(An) < 0. Define the set (1) Show that E = {xER: € An for infinitely many n}. E = {x ER: VnEN, 3no >n: x € Ang}. (2) Let En Un Ak = An U An+1 U.... Show that = EcEm VneN. (3) (a) Show that (b) Deduce that 80 7=1 m²(E) ≤ [m*(A), VEN. m* (E) = 0. (Hint: If a series converges, the remainder tends to 0)
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