I encountered the following fact:
Suppose $A$ is any set of non-negative integers which is closed under addition and has greatest common divisor 1. Then $A$ must contain all but finitely many non-negative integers.
The proof presented relied on a lemma that effectively shows that there exists an $m=m(A)$ such that for all $n\in A$ with $n\geq m$ on has that $n$ can be written as a linear combination of elements of $A$ with nonnegative coefficients. The proof of this lemma is not entirely difficult but, I was wondering if there is an especially elegant one or two line proof of the fact above?