TITLE: Proving any infinite set has a denumerable subset with the Axiom of Choice
QUESTION [5 upvotes]: Derive from the axiom of choice that any infinite set contains a denumerable subset

REPLY [2 votes]: Here is another simple proof. Countable choice is enough here:
Let $X$ be an infinite set. Let's denote $X_n=\{ Y\subset X : |Y|=n\}$. Observe that because $X$ is infinite these sets are always non-empty. Now take the set $A=\{X_n : n\in\omega\}$. Assuming the countable axiom of choice we get a choice function $f$ for $A$. Take the set $B=\bigcup ran(f)$. Of course $B\subset X$ and as a countable union of finite sets it has countable many elements.