Theorem. Alternative characterisation of directedness [dt-0014]

A subset $Y \subseteq X$ of a poset $X$ is directed iff every finite subset of $Y' \subseteq Y$ has an upper bound in $Y$ .

Proof.

⇐: Every pair of elements $x,y \in Y$ has an upper bound simply by taking the upper bound of the set $\{x,y\}.$
⇒: Given a finite set $X = \{x_1,x_2,\dots , x_n\}$ , we can show it has an upper bound by an inductive process, first taking the upper bound of $x_1$ and $x_2$ , and then taking the upper bound of that an $x_3$ and so on until we have an upper bound for the whole set. This induction works because the set $X$ is finite.