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Hi there, I need fully detailed 100% correct non-plagiarized solutions in fluent U.S.English to questions 1.3.1, 1.3.2, 1.3.7, 1.3.8; 1.3.10, and 1.4.7 in the PDF file. This needs to be done by Friday - Sep 15 at 9:00 am EST. Please do not bid if you can't get it done.
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Explanation & Answer
attached is my complete answer
Question 1.3.1:
(a) A real number s is the infimum or greatest lower bound of a set A
, i.e., s inf A , if
it meets the following criteria:
(i) s is a lower bound for A
(ii) If b is any lower bound for A then s b
(b) A version of Lemma 1.3.8 for greatest lower bounds can be stated as follow:
Lemma 1.3.8: Assume s
is a lower bound for a set A
. Then, s inf A
if and only if, for every choice of 0 , there exists an element a A such that
a s .
Proof:
Suppose s inf A and let 0 be given. Then, by definition of infimum, s is not a
lower bound of A , hence there exists a A satisfying a s .
Suppose
for all 0 , there exists an element a A such that a s . Let t be
another upper bound of A . In order to show s inf A , we have to prove s t by
contradiction. Suppose t s , and set 0 t s 0 . There exists a A satisfying
a s 0 t , which yields that t is not a lower bound of A , a contradiction.
Hence, we must have s t and it implies that s inf A , as desired.
Question 1.3.2:
(a) For a set B with inf B sup B , let x be an element in B , i.e., x B . By definition of
infimum and supremum, we have inf B x and x sup B . Combing with inf B �...