Hey davidciprut.

The proofs look good: the only suggestion I have for you is to make sure you refer to elements as members of a set (like a E A) in the formal proof.

Also as some food for thought, one of the ways that you prove stuff with natural numbers is to use the well ordering theorems where the natural numbers has a least element (usually 1 but sometimes for some definitions it can be 0). This technique is used a lot in number theory proofs and also within a lot of pure mathematics proofs that range across different areas (not just number theory).