Proving any infinite set has a denumerable subset with the Axiom of Choice Derive from the axiom of choice that any infinite set contains a denumerable subset

A somewhat simpler solution:
Suppose $X$ is an infinite set, using the well ordering principle (which is equivalent to the axiom of choice) take any well ordering of $X$, since $X$ is infinite the order type is some $\alpha>\omega$.
Now simply take the first $\omega$ elements of the order, it is a countable subset of $X$.
