Countability of Subsets proof

Prove: A subset of a countable set is countable.

I began the proof saying we will assume a set B which is a subset of A where A is countable. By the following proposition, which says,

" the nonempty set A is countable if and only if there exists a surjection N --> A ",

there exists a surjection N --> A

Then I think there exists a surjection from N --> B, which would mean B is countable as well, but I am not sure how to justify this statement. Any help with this proof would be appreciated!