Induction seems like it will be useful here, and I believe you'll need to invoke the axiom of choice during the induction step to choose an element of A which is not in your subset of size n
Some authors say a set infinite if it is not finite.
Others say a set infinite if it is equipotent to a proper subset of itself.
Still others require an infinite set to contain a copy of the natural numbers.
While any of these lead to the same set of theorems and problems, the definition you are given it makes an enormous difference in the way one writes up a solution.
So how does your textbook/instructor define infinite set?