[Math] Is the proof that the set of all finite subsets of a countable set is countable correct

Question

Q. Let $X$ be a countable, infinite set. Prove that the set of all finite subsets
of $X$ is countable.

So I say;

Let $X$ countable be given.

Let $F$ be the set of all finite subsets of $X$.

Let $F_i$ be the subsets of $X$ containing $i$ elements where $i$ a natural number.

Then $F$ is the union of all the $F_i$.

This union is countable as the natural numbers are countable.

A countable union of finite sets is countable, hence $F$ is countable.

(I have already proved a countable union of finite sets is countable in an earlier stage of the question, that is why I am stating this as fact)

Does this proof hold? I have seen a few other proofs that seem to be more complicated than mine and so I worry that I am missing something simple.

Answer 1

You say that $F$ is countable because it is a countable union of finite sets, which is false. $F_1$ is not a finite set if $X$ is infinite, and neither is $F_i$ for any other $i$!