Axiom:Axiom of Dependent Choice

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

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

The axiom of dependent choice states that there exists a sequence $b_0, b_1, b_2, \ldots \in S$ such that:
 * $\forall n \in \N: b_n \ \mathcal R \ b_{n+1}$

Also see
This axiom is a weaker form of the axiom of choice, as shown in Axiom of Choice Implies Axiom of Dependent Choice.