Euclidean Algorithm/Proof 1

Proof
Suppose $a, b \in \Z$ and $a \lor b \ne 0$.

From the Division Theorem, $a = q b + r$ where $0 \le r < \left|{b}\right|$.

From GCD with Remainder, the GCD of $a$ and $b$ is also the GCD of $b$ and $r$.

Therefore, we may search instead for $\gcd \left\{{b, r}\right\}$.

Since $\left\vert{r}\right\vert < \left\vert{b}\right\vert$ and $b \in \Z$, we will reach $r = 0$ after finitely many steps.

At this point, $\gcd \left\{{r, 0}\right\} = r$ from GCD with Zero.