Let $x,y\in\Bbb R$. We define addition operation $(+)$ by $$x+ y:=\inf\{r+ s\mid r,s\in\Bbb Q \text{ and } x<r \text{ and } y<s\}$$
Theorem: Let $x\in\Bbb R$, $a=\inf \{r \mid r\in\Bbb Q,x<r\}$, and $b=\inf\{-s\mid s\in\Bbb Q,s<x\}$. Prove that $$a+b=\inf\{r-s \mid r,s\in\Bbb Q,s<x<r\}$$
This is a theorem that the authors of my textbook Introduction to Set Theory by Hrbacek and Jech leave as an exercise. The theorem is from Chapter 10. Sets of Real Numbers. Could you please verify my attempt? Thank you for your help!
My attempt:
It is clear that $a=x$ from $a=\inf \{r \mid r\in\Bbb Q,x<r\}$.
By definition, $a+b=\inf\{r+p\mid r,p\in\Bbb Q,a<r,b<p\}=\inf\{r+p\mid r,p\in\Bbb Q,x<r,b<p\}$.
Substituting $-p$ for $p$, we get $a+b=\inf\{r-p\mid r,-p\in\Bbb Q,x<r,b<-p\}=$ $\inf\{r-p\mid r,p\in\Bbb Q,x<r,b<-p\}$.
We have $p\in\Bbb Q$ and $b<-p \iff p\in\Bbb Q$ and $-p>-s$ for some $s\in\Bbb Q$ such that $s<x$ $\iff p\in\Bbb Q$ and $p<s$ for some $s\in\Bbb Q$ such that $s<x$ $\iff p=s$ for some $s\in\Bbb Q$ such that $s<x$.
It follows that $a+b=\inf\{r-p\mid r,p\in\Bbb Q,x<r,b<-p\}=$ $\inf\{r-s\mid r,s\in\Bbb Q,x<r,s<x\}=\inf\{r-s\mid r,s\in\Bbb Q,s<x<r\}.$