Subset of A countable Set is countable

I am trying to prove that any subset of a countable set is countable.

I'm working with the with S being countably infinite right now.

Proof:

Let S be countably infinite. A set S is countably infinite iff S is equivalent to the set J of positive integers. Then there exists f such that S is 1-1 and onto J. A subset of S will also be in the set of positive integers. Thus, also countably infinite. Therefore, T is countable.

I think i went wrong somewhere. Do I need to find a 1-1 and onto function T to J?