Proving that AUB is countable Infinite given a specific case

(a) Prove: If A is a ﬁnite set and B is a countable inﬁnite set and A (intersection) B = null set, then AUB is

countable inﬁnite.

(b) Is 'A (intersection) B = null set ' a necessary assumption?

For A, I understand that I need to find a bijection f:B to Natural numbers, then prove that it's 1-1 and onto by proving its invertiblity.

However, where I get confused is how to incorporate that A is a finite set. After proving that B is countable infinite and that it has a bijection with N, then is there a way to prove that if A is finite, then AUB is also countably infinite?

Thank you.

Re: Proving that AUB is countable Infinite given a specific case