"There are those who support the idea of simply declaring that 0 (an additive identity) ..." Explain what an additive identity is, and why it makes sense to call it zero.
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Item 3: "... salvage the idea by adding a condition that the set has a lower bound." How do you think the term lower bound should be defined? If x is a lower bound for the set S, what properties should x satisfy?
Also Item 3: do you think it should be possible for a set to have more than one lower bound? Why or why not?
Item 7: "... if for no other reason than it will not be true if m is an additive identity." Explain why the property would not be true if m is an additive identity.
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Item 16: "We cannot keep the general statement in integers." Why do they say that the general statement cannot be kept if m, n, and k are integers?
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"Number 3 is just the distributive property now that number 2 is proved." Work out the details and prove statement # 3.
"Number 4 also follows from number 2 and the associative and commutative properties of multiplication." Work out the details and prove statement # 4.
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"We will add the additive inverse of k to both sides of this last inequality." Why do they add the additive inverse of k, rather than multiplying by -1?
"Now −k is an integer, and no matter what it looks like, it is positive." What do they mean by "positive," in this context? Given what we know so far, how would you define "positive" and "negative" numbers?
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"So we will prove that if m < n is true, then m + 1 ≤ n." What type of proof is therefore used to prove Claim 1?
"Assume m < n. Then 0 < n − m." Explain how the latter inequality is obtained from the former.
"So 1 ≤ n − m. So m + 1 ≤ n." Explain how to obtain the second statement from the first.