Show that, using the axiom of choice, that the cardinality of the sets of all countable subsets of $\mathbb{R}$ have cardinality $2^{\aleph_0}$ and show where it was used the axiom of choice.
This is a question of my homework. I could already prove that the cardinality of all finite subsets of $\mathbb{R}$ is $2^{\aleph_0}$ and I know that every real finite set is countable. I don't know how to use this (if I need this) and how to use de axiom of choice.