# Upper Set is Convex

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## Theorem

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

Let $T \subseteq S$ be an upper set.

Then $T$ is convex in $S$.

## Proof

Let $a, c \in T$.

Let $b \in S$.

Let $a \preceq b \preceq c$.

Since:

$a \in T$
$a \preceq b$
$T$ is an upper set

it follows that:

$b \in T$

This holds for all such $a$, $b$, and $c$.

Therefore, by definition, $T$ is convex in $S$.

$\blacksquare$