The basic definition is, A set is said to be countable if there exists one to one mapping of every element of that set with a natural number.

Now here are talking about a subsetof a countable set, it means set is already countable, in the sense there exist already mapping with a natural number of each element of this set, whatever subset we take out this set, those subsets will have also mapping with a natural number.

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

A finite set is a set with finite number of elements and hence all finite sets are countable. In addition some infinite sets are also countable if they have the same cardinality as some subset of natural numbers like the set of natural numbers, set of even numbers, etc. So, naturally any subset of a countable set must also be countable - it may or may not be finite.