If A and B are nonempty, bounded, and satisfy A c B then sup A < sup B. If sup A < inf B for sets A and B, then there exists a c e R satisfying a < c< b for all a E A and b e B. If there exists a c E R satisfying a
If A and B are nonempty, bounded, and satisfy A c B then sup A < sup B. If sup A < inf B for sets A and B, then there exists a c e R satisfying a < c< b for all a E A and b e B. If there exists a c E R satisfying a
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 10TFE
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![Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are
not:
(a)
If A and B are nonempty, bounded, and satisfy A c B then sup A < sup B.
(b)
If sup A < inf B for sets A and B, then there exists a c e R satisfying a <c < b for all a E A and b e B.
(c)
If there exists a c e R satisfying a <c < b for all a E A and b e B, then sup A < inf B.
(d)
There exist two sets A and B such that An B = 0, sup A = sup B, sup A ¢ A and sup B¢ B.
%3D
(e)
There exists a sequence of nested open intervals J Ɔ J, Ɔ J3 . with n Jn 7 Ø but containing only a finite number of
'n=1
elements.
(f)
There exists a sequence of closed (not necessarily nested) intervals I1 , I2, I3,., such that InC[-1,1] for all natural n with
...
the property that n,In # Ø for all natural numbers N, but n In = 0.
(g)
There exists a bounded sequence (an), and a sequence (b„) such that lim,→00 b, = 0, but lim,→00 ambn 7 0.
(h)
There exist a convergent series Exn and a bounded sequence (yn) such that ExnYn diverges.
(i)
There exists a convergent sequence (xn) E R" such that ||xn||
Vn.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc488e4e9-7f79-4c49-a329-792e338269fa%2F3cc19e4d-0812-4ba2-945b-8a901f1cd9db%2F8sbsgjb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are
not:
(a)
If A and B are nonempty, bounded, and satisfy A c B then sup A < sup B.
(b)
If sup A < inf B for sets A and B, then there exists a c e R satisfying a <c < b for all a E A and b e B.
(c)
If there exists a c e R satisfying a <c < b for all a E A and b e B, then sup A < inf B.
(d)
There exist two sets A and B such that An B = 0, sup A = sup B, sup A ¢ A and sup B¢ B.
%3D
(e)
There exists a sequence of nested open intervals J Ɔ J, Ɔ J3 . with n Jn 7 Ø but containing only a finite number of
'n=1
elements.
(f)
There exists a sequence of closed (not necessarily nested) intervals I1 , I2, I3,., such that InC[-1,1] for all natural n with
...
the property that n,In # Ø for all natural numbers N, but n In = 0.
(g)
There exists a bounded sequence (an), and a sequence (b„) such that lim,→00 b, = 0, but lim,→00 ambn 7 0.
(h)
There exist a convergent series Exn and a bounded sequence (yn) such that ExnYn diverges.
(i)
There exists a convergent sequence (xn) E R" such that ||xn||
Vn.
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