Let $(X,+_X,*_X,\le_X)$ be an Archimedean totally ordered field. I've already proven that by making a function from natural numbers to $X$ of the form $\phi (n \in \mathbb N)=\underbrace{1_K+...+1_K}_{added \,n\,times}$, extending it to negative integers by $\phi(z)=-\phi(-n)$ and to rationals by $\phi(\frac{a}{b})=\frac{\phi(a)}{\phi(b)}$ I get an isomorphism between $\mathbb Q$ and the set of values of $\phi:\mathbb Q \rightarrow X$. From now on I'll refer to this isomorphic copy simply as "rationals".
I've proven that ${\forall}x\ge0_X:\exists q \in \mathbb Q:q
How to prove the theorem in the title for $x<0$, however?