# Principles of Mathematical Analysis, Dedekind Cuts, Multiplicative Inverse

At the top of the page 20 of Rudin's book ''Principles of Mathematical Analysis'' he writes: ''The proofs (of the multiplication axioms) are so similar to the ones given in detail in Step 4 (proof of the addition axioms) that we omit them''. I tried to prove them but I got stuck in the proof of $$\alpha \cdot {\alpha }^{-1}=1^*$$ where $\alpha$ is positive cut and ${\alpha }^{-1}=\mathbb{Q}_{-}\bigcup\left\{0\right\}\bigcup\left\{t\in \mathbb{Q}:0<t<r\text{ for some }r\in \mathbb{Q}:\frac{1}{r}\notin \alpha\right\}$ is the candidate for the multiplicative inverse of $\alpha$. I have already proved that ${\alpha }^{-1}$ is a cut and $\alpha \cdot {\alpha }^{-1}\le 1^*$.

My question is how do we prove the opposite direction similarly to the proof Rudin gives for $\alpha +(-\alpha) \le 0^*$. A proof completely different to that one can be found here: Dedekind cut multiplicative inverse

Here is what I have tried thus far:

Let $p\in 1^*$. If $p\le 0$ then obviously $p\in \alpha\cdot \alpha^{-1}$.

Suppose $0<p<1$ and $q=q(p)\in \mathbb{Q}_{+}$. By the Archimedean Property of Rational numbers $$\exists n\in \mathbb{N}:nq\in \alpha\text{ and }(n+1)q\notin \alpha$$ We must find a $u \in \alpha^{-1}$ such as that $p=(nq)\cdot u$ or equivalenty, $u=\frac{p}{nq}$

In order for $u \in \alpha^{-1}$ we must have that $0<u<r$ and $\frac{1}{r}\notin \alpha$ for some rational $r$. The only reasonable choice for $r$ would be $\frac{1}{(n+1)q}$. But then, $$u<r\Leftrightarrow \frac{p}{nq}<\frac{1}{(n+1)q}\Leftrightarrow p<\frac{n}{n+1}$$ which may not be true for some values of $n$ (like $0$). Where can we derive a restriction for these values of $n$?

EDIT: Found another proof here: http://mypage.iu.edu/~sgautam/m413.33418.11f/Dedekind.pdf STill nothing similar to Rudin's...

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Let $p\in 1^*$, $0 < p < 1$, it exists $n\in \mathbb N$ such that $$p < 1 - \frac 1 {m + 1} = \frac m {m + 1} \tag{1}$$ for each $m\in \mathbb N$, $m \geq n$.

Let $r\in \alpha, r >0$ and $0 < q < r/n$. There exists a $m$ such that $m q\in \alpha$ and $(m + 1)q\notin \alpha$. Evidently we have $m \geq n$.

Inequality (1) implies $$\frac p {mq} < \frac m {m + 1}\cdot \frac 1 {mq} = \frac 1 {(m + 1) q}$$ so $\frac p {mq} \in \alpha^{-1}$ and $$p = (mq)\cdot \frac p {mq}$$

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Very very nice proof! Not exactly what I was looking for but still, this is fantastic! You made my day! Note: In the beggining you must let $0<p<1$, but that won't affect proof at all as the other case is obvious – Nameless Jul 27 '12 at 13:29
I assumed $0 < p < 1$, because that was the only case you had difficulties with. Anyway, now, I explicitly mentioned that condition. – AlbertH Jul 28 '12 at 11:22
Sorry to reopen a question decided two years ago, but precisely why is it evident that $m \geq n$? Otherwise I agree completely... – User12345 Aug 31 '14 at 23:52
Since $p < 1$, there exists an integer $n$ great enough to satisfy $p < 1 - (n + 1)^{-1}$. Then for each $m$ greater than $n$ inequality (1) holds. – AlbertH Sep 1 '14 at 8:34
@User12345 $0<q<r/n$ and $mq<rm/n$ then $(m+1)q<r(m+1)/n$. If $m<n$ then $m=n-k$ for some $k\geq 1$ hence $(m+1)/n = (n-k+1)/n = 1+(1-k)/n\leq 1$. This implies $(m+1)q<r(m+1)/n\leq r$, then $(m+1)q\in\alpha$, which is contradictory. – Fonseca Jan 28 at 16:02

suppose $m<n$, then $m\le n-1$ and $m+1 \le n$. As a result,$(m+1)q\le nq<r$ and hence $(m+1)q$ belongs to $\alpha$, which contradicts the choice of $m$.

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This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. – Joonas Ilmavirta Sep 19 '14 at 8:02
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. – Joonas Ilmavirta Sep 19 '14 at 8:02