Intersection of Convex Sets is Convex Set (Order Theory)

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $\mathcal C$ be a set of convex sets in $S$.

Then $\displaystyle \bigcap \mathcal C$ is convex.

Proof
Let $a, b, c \in S$.

Let $a, c \in \displaystyle \bigcap \mathcal C$.

Let $a \prec b \prec c$.

By the definition of intersection:


 * $\forall C \in \mathcal C$: $a, c \in C$

Since each $C \in C$ is convex:


 * $\forall C \in \mathcal C$: $b \in C$.

Thus by the definition of intersection:


 * $b \in \displaystyle \bigcap \mathcal C$

Thus $\displaystyle \bigcap \mathcal C$ is convex.