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[0, 1]. Prove that x e C iffx has a ternary expansion with a, e {0,2} for 1. Let I be the interval [0, 1]. Remove the open middle third segment (.2) ana let Aj be the set that remains. That is, A1= Then remove the open middle third segment from each of the two parts of A, and call the remaining set A2. Thus A2 = Continue in this manner. That is, given Ak, remove the open middle third segment from each of the closed segments whose union is A, and call the remaining set Ak+1. Note that A1 2 A2 2 A3 2 ·... and that for each k e N, Ak is the union of 2* closed intervals each of length 3 . The set C = 4 is called the Cantor set. k%3D1 (a) Prove that C is compact. (b) Let x = 0.aja2az •*… be the ternary (base 3) expansion of a number ... all n e N. (c) Prove that C is uncountable. (d) Prove that C contains no intervals.

[0, 1]. Prove that x e C iffx has a ternary expansion with a, e {0,2} for 1. Let I be the interval [0, 1]. Remove the open middle third segment (.2) ana let Aj be the set that remains. That is, A1= Then remove the open middle third segment from each of the two parts of A, and call the remaining set A2. Thus A2 = Continue in this manner. That is, given Ak, remove the open middle third segment from each of the closed segments whose union is A, and call the remaining set Ak+1. Note that A1 2 A2 2 A3 2 ·... and that for each k e N, Ak is the union of 2* closed intervals each of length 3 . The set C = 4 is called the Cantor set. k%3D1 (a) Prove that C is compact. (b) Let x = 0.aja2az •*… be the ternary (base 3) expansion of a number ... all n e N. (c) Prove that C is uncountable. (d) Prove that C contains no intervals.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 74E
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Can you solve this #11 please. Thank you.

0.aja2a3 * be the ternary (base 3) expansion of a number x in [0, 1]. Prove that x e C iffx has a ternary expansion with a, e {0,2} for (e) Prove that e C but is not an endpoint of any of the intervals 11. Let I be the interval [0, 1]. Remove the open middle third segment (! 2) Jon al let A, be the set that remains. That is, A1 = 0, 3 %3D Then remove the open middle third segment from each of the two parts of A. and call the remaining set A2. Thus 11, [2 1] 0, 2 7 A2 edua lor Continue in this manner. That is, given Ak, remove the open middle third segment from each of the closed segments whose union is Ak, and call the remaining set Ak+1• Note that A1 2 A2 2 Az =... and that for each ke N, Ak is the union of 2 closed intervals each of length 3-k. The set C = 4 is called the Cantor set. k=1 =D1 (a) Prove that C is compact. (b) Let x = 0.ajazaz •…· [0, 1]. Prove that x e C iff x has a ternary expansion with a, e {0,25 e dot all n e N. (c) Prove that C is uncountable. %3D be the ternary (base 3) expansion of a number x n ... olb (d) Prove that C contains no intervals. (e) Prove (6) Prove that EC but is not an endpoint of any of the intervas in any of the sets A (k e N)
Transcribed Image Text:0.aja2a3 * be the ternary (base 3) expansion of a number x in [0, 1]. Prove that x e C iffx has a ternary expansion with a, e {0,2} for (e) Prove that e C but is not an endpoint of any of the intervals 11. Let I be the interval [0, 1]. Remove the open middle third segment (! 2) Jon al let A, be the set that remains. That is, A1 = 0, 3 %3D Then remove the open middle third segment from each of the two parts of A. and call the remaining set A2. Thus 11, [2 1] 0, 2 7 A2 edua lor Continue in this manner. That is, given Ak, remove the open middle third segment from each of the closed segments whose union is Ak, and call the remaining set Ak+1• Note that A1 2 A2 2 Az =... and that for each ke N, Ak is the union of 2 closed intervals each of length 3-k. The set C = 4 is called the Cantor set. k=1 =D1 (a) Prove that C is compact. (b) Let x = 0.ajazaz •…· [0, 1]. Prove that x e C iff x has a ternary expansion with a, e {0,25 e dot all n e N. (c) Prove that C is uncountable. %3D be the ternary (base 3) expansion of a number x n ... olb (d) Prove that C contains no intervals. (e) Prove (6) Prove that EC but is not an endpoint of any of the intervas in any of the sets A (k e N)
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