Directed iff Finite Subsets have Upper Bounds

Theorem
Let $\left({S, \precsim}\right)$ be a preordered set.

Let $H$ be a non-empty subset of $S$.

Then $H$ is directed


 * for every a finite subset $A$ of $H$
 * $\exists h \in H: \forall a \in A: a \precsim h$

Sufficient Condition
Let $R$ be directed.

We will prove by induction of the cardinality of finite subset of $H$.

Base case

 * $\forall A \subseteq H: \left\vert{A}\right\vert = 0 \implies \exists h \in H: \forall a \in A: a \precsim h$

Let $A \subseteq H$ such that
 * $\left\vert{A}\right\vert = 0$

By Cardinality of Empty Set:
 * $A = \varnothing$

By definition of empty set:
 * $\exists h: h \in H$

Thus by definition of empty set:
 * $\exists h \in H: \forall a \in A: a \precsim h$

Induction Hypothesis

 * $\forall A \subseteq H: \left\vert{A}\right\vert = n \implies \exists h \in H: \forall a \in A: a \precsim h$