How to prove this or where can I read the proof of this fact?

Question

Mathematics

The set of all finite subsets of a countable set is countable

Answer 1

As always with countable sets, all elements (subsets) must be renumbered. For example, the binary number system is suitable here. Number all the elements of the set. Then, for each subset, write out 1 for the bits corresponding to elements in the subset and 0 for elements not in it. You get a binary number. Convert to decimal and get the subset number.
So you can get a unique subset by number, and any subset has some number. All. So we numbered the set of all subsets by natural numbers. It is countable.