Interval of Ordered Set is Convex

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Theorem

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

Let $I$ be an interval: be it open, closed, or half-open in $S$.


Then $I$ is convex in $S$.


Proof

Any interval can be represented as the intersection of two rays.



Thus by Ray is Convex and Intersection of Convex Sets is Convex Set (Order Theory), $I$ is convex.

$\blacksquare$