So the order axioms are the following:
For all $a, b, c \in S$ $$(O1) a < b \text{ or } a = b \text{ or } a>b$$ $$(O2) a<b \text{ and } b<c \therefore a<c$$ $$(O3) a < b \to a+c<b+c$$ $$(O4) a<b \text{ and } c>0 \therefore ac < bc$$
Now, since all Cauchy sequences are convergent sequences, let $[a_k], [b_k], [c_k]$ be the equivalence classes of the Cauchy sequences that converge to $a,b,c$ respectively. I mean, since $a,b,c \in \mathbb{R}$ and the reals are an ordered field, the order axioms follow automatically. But is this enough?
EDIT: The definition I am using is $a<b$ if $a_k<b_k$ eventually in a Cauchy sequence and viceversa with $a>b$.