A subset of positive integers closed under addition with gcd =1 must have all but a finite number of the non negative integers

Question

I have done no number theory. With that in mind, please give me a proof for the statement below:

$(A\subseteq\Bbb Z^+) \land(x\in A \land y\in A\implies x+y\in A)\land (gcd(A)=1) \implies \exists n_0\in \Bbb Z^+,\forall n\in\Bbb Z^+,n\ge n_0(n \in A) $

I have seen other proofs like this

and this

which I wasn't able to follow. Whatever terms you're using here please define them. Please don't presuppose of me any knowledge of number theory.