Axiom:Axiom of Dependent Choice/Left-Total

Axiom
Let $\mathcal R$ be a binary relation on a non-empty set $S$.

Suppose that:
 * $\forall a \in S: \exists b \in S: a \mathrel{\mathcal R} b$

that is, that $\mathcal R$ is a left-total relation.

The axiom of dependent choice states that there exists a sequence $\left\langle{x_n}\right\rangle_{n \in \N}$ in $S$ such that:
 * $\forall n \in \N: x_n \mathrel{\mathcal R} x_{n+1}$