Prove that for fixed , there is exactly one such that .

First, we prove a lemma.

Lemma: 1 is the smallest element of .

* Proof. * Consider the set . Then, is an inductive set since and for all , . Thus, (since, by definition, is the set of elements common to every inductive set). But, for any , if , then and hence . Thus, 1 is the least element of .

Now, for the theorem of the exercise.

* Proof. * First, we prove existence.

Let . We know is non-empty since by I.3.12, Exercise #2 we know there exist integers and such that . This must be in our set . We also know is bounded above since is an upper bound by our definition of . Therefore, by the least upper bound axiom (Axiom 10, p. 25), we know exists.

Then, by Theorem I.32, we know that for any positive real number , there is some such that

Letting , we have

Therefore, since the supremum of is an upper bound, we know ; therefore, by definition of . But we already know , and so by definition of we must have . Therefore, we have found an integer such that

Now, for uniqueness. Suppose there are integers with this property. Then we have

Then, adding the terms of these inequalities we have,

Without loss of generality, assume . Then this inequality implies, . But since 1 is the smallest element of (from the lemma above) and is closed under subtraction, we must have , i.e., . Thus, any such is unique

You missed a prime off one of your n primes. Maybe using m instead of n prime would make this proof more readable.

The statement `Then, by Theorem I.32, we know that for any positive real number h, there is some n in S such that n > sup S – h` is incorrect since S only contains integers and is not dense. consider the following example. take x = 4.7 and h = 0.1. sup S – h = 4.6. No n bigger than 4.6 exists

I think it’s also unnecessary, since if is any other integer less than , then , so .

It’s also possible to eliminate the appeal to the lemma: suppose is another integer such that , then , so , contradicting the hypothesis. Therefore, and , as required.

Hmm, the text of my comment got a little truncated. Here’s what I wanted to say:

Suppose is another integer such that . Then , so (since is the supremum). Hence, ; set (such a exists by Theorem I.2). Then . If , then , so , contradicting the hypothesis. So and .