I am trying to understand a theorem that proves that the supremum axiom, Dedekind axiom, and continuity axiom are all equivalent. I have trouble understanding one point in the proof that DED implies CON. So for the definition, the book I'm using defines the following:
$(1)$ Let X be an ordered set. The pair $(D_1, D_2)$ of subsets $D_1, D_2 \subset X$ is called a Dedekind cut if it has the following properties: $D_1 \cup D_2 = X\tag{A}$ $D_1 \neq \emptyset, \ D_2 \neq \emptyset\tag{B}$ $\delta_1 \in D_1 \wedge \delta_2 \in D_2 \implies \delta_1 < \delta_2\tag{C}$
The axioms are stated as follows:
(DED) If $D_1$, $D_2$ $\subset \mathbb{R}$ are such that $(D_1, D_2)$ is a Dedekind cut then there exists a least upper bound sup $D_1 \in \mathbb{R}$ of the set $D_1$.
(CON) Let $T$ and $S$ be two nonempty subsets of $\mathbb{R}$ with the property: $(\forall t\in T)(\forall s\in S) t\leq s$. Then there exists $c\in\mathbb{R}$ such that $(\forall t\in T)(\forall s\in S)t\leq c \leq s$.
Finally, the proof (or the part that is weird) goes as follows (keep in mind the implication we are trying to prove, namely DED $\implies$ CON):
Suppose DED holds and let $T, S$ be sets with the property from CON. Let $D_1 = \{x\in\mathbb{R}\rvert (\forall s\in S)x<s \}$ and $D_2 = \mathbb{R}\setminus D_1$. Then, $(D_1, D_2)$ is a Dedekind cut. To prove this, first note that $(A)$ holds from the definition of sets $D_1, D_2$. Let $\epsilon > 0$ and $$ T_{\epsilon} := \{t-\epsilon\lvert t\in T\} $$
Obiously $S\subset D_2$ and $T_\epsilon \subset D_1$, since by assumption of CON, $T$ and $S$ are nonempty we proved (B). If $\delta_1 \in D_1$, then $\delta_1<s$ for all $s\in S$. Let: $$ \delta_2\in D_2. $$ If it were to be $\delta_2 \leq \delta_1$, then it would imply $\delta_2 < s$ for all $s\in S$. By definition of $D_1$ this would yield $\delta_2 \in D_1$,which together with $\delta_2\in D_2$ and (C) gives the contradiction $\delta_2<\delta_2$. Thus $(D_1, D_2)$ is a Dedekind cut.....(he goes on to prove the rest but I understand that part)
The only problem I have is with the very last bit. How do we suppose (C)? Aren't we trying to prove (C)? If someone can clarify the argument above or suggest an improvement I would be very grateful.