Exercise 3.3.2. Decide which of the following sets are compact. For those that are not compact, show how Definition 3.3.1 breaks down. In other words, give an example of a sequence contained in the given set that does not possess a subsequence converging to a limit in the set. (a) N. (b) Qn [0,1]. (c) The Cantor set. (d) {1+1/22+1/32 +...+1/n²:ne N}. (e) {1, 1/2,2/3,3/4, 4/5,...}.
Exercise 3.3.2. Decide which of the following sets are compact. For those that are not compact, show how Definition 3.3.1 breaks down. In other words, give an example of a sequence contained in the given set that does not possess a subsequence converging to a limit in the set. (a) N. (b) Qn [0,1]. (c) The Cantor set. (d) {1+1/22+1/32 +...+1/n²:ne N}. (e) {1, 1/2,2/3,3/4, 4/5,...}.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...
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![Exercise 3.3.2. Decide which of the following sets are compact. For those that
are not compact, show how Definition 3.3.1 breaks down. In other words, give
an example of a sequence contained in the given set that does not possess a
subsequence converging to a limit in the set.
(a) N.
(b) Qn [0,1].
(c) The Cantor set.
(d) {1+1/22+1/32 +...+1/n²:ne N}.
(e) {1, 1/2,2/3,3/4, 4/5,...}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6389447-1237-4af0-b5c6-eb1260425b55%2F55191bd5-7444-4f17-9446-d605960acbd8%2F7wbbho_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 3.3.2. Decide which of the following sets are compact. For those that
are not compact, show how Definition 3.3.1 breaks down. In other words, give
an example of a sequence contained in the given set that does not possess a
subsequence converging to a limit in the set.
(a) N.
(b) Qn [0,1].
(c) The Cantor set.
(d) {1+1/22+1/32 +...+1/n²:ne N}.
(e) {1, 1/2,2/3,3/4, 4/5,...}.
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