$a)$ the set of all rational numbers is countable;
$b)$ the set of all real numbers is uncountable;
$c)$ any subset of a countable set is countable;
$d)$ the union of countably many countable sets is countable.
I have no idea how to formally prove this.
For $a)$, I think, the idea would be to notice that any rational number can be expressed like fraction. Taking the number of the denominator $n$ for establishing a relation to set of integers there would be $n-1$ numbers for each integer. So it would be a union of countably many countable sets. But then I would need to prove $d)$ first.
Could anyone show me an example of this type of proofs or give any advice.