Let $K$ be a field and $\mathcal{O}$ be a valuation ring of $K$ (So $\mathcal{O}\subset K$ is an integral domain with the property that either $x$ or $x^{-1}$ is in $\mathcal{O}$ for all $x\in K$). Define a valuation $\nu: K^\times\rightarrow K^\times/\mathcal{O}^\times$ by the quotient map, where $\cdot^\times$ denotes the group of units.
Now for elements $a,b\in K^\times$ we define for $a\mathcal{O}^\times$ and $b\mathcal{O}^\times$ that $a\mathcal{O}^\times \le b\mathcal{O}^\times$ if $a^{-1}b\in\mathcal{O}$. Goldschmidt in 'Algebraic functions and projective curves' claims that this defines a total ordening on $K^\times/\mathcal{O}^\times$, but I do not fully understand this. Say that $a,b \in \mathcal{O}$ and $a^{-1},b^{-1}\notin \mathcal{O}$, then I don't understand why either $a^{-1}b$ or $b^{-1}a$ has to be in $\mathcal{O}$.
I assume it is true, but can someone explain to me why this is always the case? Thank you!
Solved in comments:
Just take $a^{-1} b$ to be the element $x$ in the definition of a valuation ring.