Integer Combination of Coprime Integers/Necessary Condition/Proof 2

Proof
Let $a, b \in \Z$ be such that $\exists m, n \in \Z: m a + n b = 1$.

Let $d$ be a divisor of both $a$ and $b$.

Then:
 * $d \divides m a + n b$

and so:
 * $d \divides 1$

Thus:
 * $d = \pm 1$

and so:
 * $\gcd \set {a, b} = 1$

Thus, by definition, $a$ and $b$ are coprime.