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## Main Question or Discussion Point

The Axiom of Dependent Choice, a weaker version of the Axiom of Choice, states that for any nonempty set X and any entire binary relation R on X, there is a sequence (x_n) in X such that x_n R x_n+1 for each n in N.

My question is, what would happen if you restricted the relations to functions, i.e. for any nonempty set X and any function f from X to X, there is a sequence (x_n) in X such that f(x_n) = x_n+1 for each n in N? Would that be equivalent to the Axiom of Dependent Choice, or would it be weaker? If it's weaker, how weak is it? Is it provable in ZF?

Any help would be greatly appreciated.

Thank You in Advance.

My question is, what would happen if you restricted the relations to functions, i.e. for any nonempty set X and any function f from X to X, there is a sequence (x_n) in X such that f(x_n) = x_n+1 for each n in N? Would that be equivalent to the Axiom of Dependent Choice, or would it be weaker? If it's weaker, how weak is it? Is it provable in ZF?

Any help would be greatly appreciated.

Thank You in Advance.