That's not even true! A subset of a countable set may be finite! Your statement "A subset of S will also be in the set of positive integers" is also not true- it is not assumed that S itself is "in" the set of positive integers. You may have meant that there is a 1-1 and onto function from the subset to positive integers but that is exactly what you want to prove.- you can't just say it!

You mean, I presume, "anyI think i went wrong somewhere. Do I need to find a 1-1 and onto function T to J?infinitesubset of a countable set is countably infinite". What you need to do is show that there exist 1-1 onto function from the subset to N (the standard notation of the set of positive integers. There is no need for a "J".)