Union of Overlapping Convex Sets in Toset is Convex/Infinite Union

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

Let $\mathcal A$ be a set of convex subsets of $S$.

Suppose that for any $A, B \in \mathcal A$, there are elements $C_0, \dots, C_n \in \mathcal A$ such that:
 * $C_0 = A$
 * $C_n = B$
 * For $k=0, \dots, n-1$: $C_k \cap C_{k+1} \ne \varnothing$

Then $\bigcup \mathcal A$ is convex in $S$.

Proof
Let $a, c \in \bigcup \mathcal A$.

Let $b \in S$.

Let $a \prec b \prec c$.

Since $a, c \in \bigcup \mathcal A$, there are $P, Q \in A$ such that $a \in P$ and $c \in Q$.