This question refers to the construction of R from Q using Dedekind cuts, as presented in Rudin's "Principles of Mathematical Analysis" pp. 17-21.
Specifically, I cannot proof (b) in step 8, bottom of pp.20. To be more precise, I'm not able to show that ${(rs)^*\subset r^*s^*}$ when $r>0$ and $s>0$, could somebody prove it for me?
Here are the original texts in Baby Rudin.
Step 8 We associate with each $r\in Q$ the set $r^*$ which consists of all $p\in Q$ such that $p < r$. It is clear that each $r^*$ is a cut; that is, $r^* \in R$. These cuts satisfy the following relations :
(a) $ r^ * + s^* = (r + s)^*$,
(b) $r^*s^* = (rs)^*$,
(c) $r^* < s^*$ if and only if $r < s$.