Prove that all subsets of countable sets are countable

This is basically a problem of my assignment, it says...

A set is said to be countable if it is either finite or there is an enumeration(list) of the set. Then prove that All subsets of countable sets are countable.

While proving the above, do I have to prove all these 3 cases separately?

• Finite subset of a finite set is countable.

• Finite subset of a countably infinite set is countable.

• Infinite subset of a countably infinite set is countable.

This is my confusion. And if we want to prove that a set $$S$$ is countable, we have to show that the function $$f:\mathbb{N}\rightarrow S$$ is a bijection. Using it, I can prove the 3rd part(infinite subset of countably infinite set); but how can I use this definition to prove the 1st two? Or, if I show that a finite set is countable, will that be enough to say that the 1st two are true?

• According to your definition, a finite set is countable. Thus there are countable sets for which there can be no bijection with $\mathbb N$. – lulu May 23 '19 at 15:56