There is some questions about connected set.

The first question arises in logical translation. I translated the property of interval into logical proposition that $\forall a,b \in E, \forall c : [a<c<b \rightarrow c \in E]$. 

Is it right? I think it is something wrong. If there is $d$ which satisfies $a<b<d$ or is not able to order $\{a,c,d\}$, then the assumption of logical proposition that $a<d<b$ is false so the statement is vacously true. What is good translation of the property of interval?

The second arises in proving "Connected set is interval". The following is definition of disconnected set, and connected set is defined by not being disconnected.

\begin{align}
E \subseteq \mathbb{R} \textrm{ is disconnected }\quad \Leftrightarrow \quad & \exists A,B \textrm{ which are open sets such that } \\ 
&(i) A \cap E \not = \varnothing, B \cap E \not = \varnothing \\
&(ii) (A \cap E) \cap (B \cap E) = \varnothing \\
&(iii) (A \cap E) \cup (B \cap E) = E
\end{align}

The following is the proof of "if $E \subseteq \mathbb{R}$ is interval, then $E$ is connected set."

> 
> Suppose not. Since $A \cap E \not = \varnothing, B \cap E \not = \varnothing$, there exists $a \in A \cap E$, $b \in B \cap E $. By trichotomy property , $a=b, a<b, a>b$. We defined $A \cap E, B \cap E$ disjoint, so $a \not = b$. WLOG suppose $a<b$. Let $S = A \cap [a,b]$. For $S \subseteq [a,b]$, S is also bounded above, and there exists supremum S such that $c:=\sup S, a<c \leq b$.
> 
> 
> Claim : $c \not = b$.
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> 
> Suppose not. Then $c$ should be a interior point of $B$.($\because B$ is open set). Then there exists something suitable open ball with its center $c$ and radius $\epsilon >0$ : $B\_\epsilon (c) \subseteq B$. Then $c- \epsilon/2$ is also upper bound of $S$, so contradiction.
> 
> 
> 

I have a question in the last paragraph of this proof. How can we verify $c- \epsilon /2$ is also upper bound? What I think is to make it sense, the premise should be proved that $\forall a \in A \cup E, \forall b \in B \cup E \rightarrow a < b$. However my text doesn't prove it and just use this premise implicitly. Explain me about this.