I want to prove that if $A\subset \mathbb{R}$ is an interval then $A$ is connected.
I found this proof, and I don't understand it essentially the ii)
Suppose that $A$ is an interval but not connected, then there existe two open sub-sets $B$ and $C$ such that $A=B\cup C$ and $B\cap C = \varnothing$.
Let $x\in B$ and $y\in C$ such that $x<y$.
The bounded set $B\cap[x,y]$ has a supremum witch we denote by $z$.
i) If $z\in B,$ we have that $z<y,$ there exists $\delta <0$ such that the interval $[z,z+\delta[\subset B\cap [x,y]$, which is a contradiction.
ii) If $z\in C$ there exists an interval $]z-\delta,z], (\delta>0)$ in $C\cap [x,y]$, which is a contradiction.
Then $z\notin B$ and $z\notin C$, which is impossible because $[x,y]\subset A$. It means that $A$ is connected.
I don't understand ii). Please help me.
Thank you.