If we're working in the context of proving that the subset of any countable set is countable itself, would it not suffice to prove that there is an injection from any infinite subset of

to

itself?

This would give us the result that the subset itself is countable, however not that it's cardinality is the same as that of

. A proper injection from

to

would be:

therefore

...

Are you allowed to make the assumption that

is the smallest infinite cardinal number? If so, this would also give us the result that

, otherwise, I would need to think of another way to prove equivalence.