How to prove $\gcd(a, b) = ax + by$

Question

Let $a, b \in \mathbb N$, assume they are not both $0$. Define $L = \{n\in\mathbb N^+ \mid \exists x, y \in \mathbb{Z}: n = ax + by\}$

how do I prove the following claim:

$L$ has a minimum element $m$, i.e. $m$ is no larger than any other element of $L$

(without assuming that $\gcd(a, b) = ax + by$)

Answer 1

Hint:

$\mathbf N^*$ satisfies the well-ordering property.

Answer 2

There is almost nothing to prove (except if you are redefining the set of natural numbers $\mathbb{N}$).

Because otherwise, by the well-ordering property of $\mathbb{N}$, you just have to prove that your set is not empty.