Given two sets of real number the intersection between this two sets is an interval?
Basically this is the qusstion and I've tried to demonstrate as follow:
Let $A \subseteq \Bbb R$, $A\ne \emptyset$ and $B \subseteq \Bbb R$, $B\ne \emptyset$.
Let $C$ be the intersection between $A$ and $B$: $A \cap B = C$.
If $C > 1 \exists a \in C $ and $\exists b \in C$ and given the condition $a < b$ thanks to Completeness of the real number it's possible says that $\exists c \in C$ such that $a \le c \le b$. So we can say that $C$ could be an interval.
Is it correct or I have to add other condition?
If $a,b\in C$ then $a,b\in A$. Since $A$ is an interval, for every $c$ with $a\le c \le b$ we have $c\in A$. Repeat the reasoning and you also have $c\in B$. Then $c\in C$ thus $C$ is an interval.