A descending chain and choice axiom.

Question

Let $R$ be a relation on a set $A$ such that for every $x\in A$ there exist a element $y\in A$ such that $x\mathrel{R}y$ the there exist a function $f\colon\omega\rightarrow A$ such that $f(n^+)Rf(n)$.

I would like some hint to this exercise using axiom of choice.

Thanks!

Answer 1

HINT: Fix a choice function $G$ on non-empty subsets of $A$, and some $x_0$ in $A$, and by induction define $f(0)=x_0$, and $f(n+1)$ to be a chosen element from the relevant set of witnesses.