Chain is Directed

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

Let $C$ be a non-empty chain of $S$.

Then $C$ is directed.

Proof
Let $x, y \in C$.

By definition of connected relation:
 * $x \preceq y$ or $y \preceq x$

WLOG: suppose that
 * $x \preceq y$

Define $z = y$.

Thus by definition of reflexivity
 * $x \preceq z$ and $y \preceq z$

Hence $C$ is directed.