The set of finite unions of intervals with rational endpoints is countable.

Question

I don't know how to prove the following:

Let

$K:=\lbrace G : G$ is a union of finitely many intervals with rational endpoints$\rbrace$.

Prove that $K$ is countably infinite.

Here is my approach:

The set of intervals with rational endpoints is countably infinite as there is a bijection between this set and $\mathbb{Q}\times \mathbb{Q}$. However, I don't know how to continue.

I really appreciate any help you can provide.

Answer 1

Hint. Let $E$ be the set of intervals with rational endpoints. You already proved it is a countably infinite set. You need now to prove that the set $U$ of finite unions of elements of $E$ is also countably infinite. Clearly $U = \bigcup_{n \geq 0} U_n$, where $U_n$ is the set of union of $n$ elements of $E$. Since a countable union of countable sets is countable, it remains to show that each set $U_n$ is countable. Can you prove that last part?