Show that, using the axiom of choice, that the cardinality of the sets of all countable subsets of $\mathbb{R}$ have cardinality $2^{\aleph_0}$

Question

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.

Answer 1

Here is a general road map:

1. Show that $\Bbb{R^N}$, all the sequences of real numbers has cardinality $2^{\aleph_0}$.

2. Find a surjection from $\Bbb{R^N}$ onto the set of all countable subsets of $\Bbb R$.

3. Conclude (using what you already know) that the cardinals are equal.

Let me give you an additional hint that the axiom of choice is used in the third step. And let me also add that you cannot prove this equality of cardinals without using some of the axiom of choice.