inverses exist in Zn, and then from another conjecture about the existence of mu
ID: 1890859 • Letter: I
Question
inverses exist in Zn, and then from another conjecture about the existence of multiplicative inverses. Write a formula definition in the style of Definition 1.3.2 for the infimum or greatest lower bound of a set. Now, state and prove a version of Lemma 1.3.7 for greatest lower bounds. Let A be bounded below, and define B = {b R : b is a lower bound for A}. Show that sup B = inf A. Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Propose another way to use the Axiom of Completeness to prove that sets bounded below have greatest lower bounds.Explanation / Answer
Paul Sally's book, eh? Since you're in such a high math class, I'm not going to solve these problems for you - just give you hints. I trust that you can figure this out. :)
The axiom of completeness is this - that every nonempty set of real numbers that is bounded above has a least upper bound.
(a) To prove this, begin by nothing that sup{a} = -inf{a} (why is this true? something to think about!)
First assume that {a} has a sup and prove it has an inf
Then assume {a} has an inf and prove it has a sup
using the fact that sup{a} = -inf{a} and vice versa
(b) Think about the relationship between an infinite set (like that of all integers or natural numbers), the inf, the greatest lower bound, and simply the lower bound. This is essentially your answer - just in more mathematical wording.
(c) If "Every nonempty set of real numbers that is bounded above has a least upper bound," then the negative - what we did in (a) - is "every nonempty subset of real numbers that is bounded below has a greatest lower bound"
I hope that you found this answer useful towards your studies. It took a considerable amount of thought, time, and effort to compose, and and I'd sincerely appreciate a lifesaver rating! It would really make my day, and will allow me to continue answering your questions :)
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