Bézout's Identity/Proof 3

Proof
First we establish that:


 * $\exists x,y \in \Z: ax + by = \gcd \left\{a,b\right\}$

include George E. Andrews: Number Theory coll.2-1

Now to show that $\gcd \left\{a,b\right\}$ is the smallest positive number to satisfy the equation.

We first show that:


 * $\forall x \in \Z: \exists m,n \in \Z: x = mx + ny \ne 0 \implies d \backslash x$

include George E. Andrews: Number Theory coll.2-2