Interval of Ordered Set is Convex
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Obvious though it is, the above needs to be stated as a theorem in its own right.
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Theorem
Let $\struct {S, \preceq}$ 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.
Obvious though it is, the above needs to be stated as a theorem in its own right.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
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Thus by Ray is Convex and Intersection of Convex Sets is Convex Set (Order Theory), $I$ is convex.
$\blacksquare$