Singleton is Convex Set (Order Theory)

Theorem

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

Let $x \in S$.

Then the singleton $\left\{{x}\right\}$ is convex.

Proof

Let:

$a, c \in \left\{{x}\right\}$

$b \in S$

$a \preceq b \preceq c$

Then $a = c = x$.

Thus $x \preceq b \preceq x$.

Since $\preceq$ is a ordering, it is antisymmetric.

Thus $b = x$, so $b \in \left\{{x}\right\}$.

Since this holds for the only such triple, $\left\{{x}\right\}$ is convex.

$\blacksquare$