Positivity in the proof of the Bezout gcd identity in Apostol's book Theorem 1.2 of Apostol Analytic Number Theory about common divisor I don't know why but the link needs to be refreshed to see the content
"Theorem 1.2 Each pair of integers a and b has a common divisor d of the form d = ax + by where x and y are integers. Moreover, every common divisor of a and b divides this d. "
His proof seems doesn't depend on y and x can be negative. So if we add a condition that $y \ge x \ge 0$ in the theorem, the proof still works. However, the result will not be true. I am wondering if I miss some points in author's proof?
Also he state this theorem before the greatest common divisor. And it should prove the existence of the GCD since every common divisor divides d. So I think he doesn't assume the GCD exists.
Thanks
 A: If you examine the inductive step in Apostol's proof you will see that it does not generally lift-up any positivity of the coefficients. Namely, to get a linear common divisor of $\rm\,a,b\,$ it first obtains, by induction, a common divisor $\rm\,d\,$ of $\rm\, a-b,b\,$ of sought linear form $\rm\, d = (a-b)x+by.\,$ Since $\rm\,d\mid a-b,b\,\Rightarrow d\mid (a-b)+b = a,\,$ we infer that $\rm\,d\mid a,b,\,$ i.e. $\rm\,d\,$ is a common divisor of $\rm\,a,b.\,$ Rewriting it yields the desired form: $\rm\,d = (a−b)x+by=ax+b(y−x),\,$ i.e. linear in $\rm\,a,b.$ But this rewriting does not preserve coeff positivity, i.e. from $\rm\,x,y\ge 0\,$ we cannot infer $\rm\,x,y-x\ge 0.\,$ 
Remark $\ $ Readers familiar with the subtractive form of the Euclidean algorithm will note the analogy with the idncution step of the classical proof of the extended Euclidean algorithm 
$$\rm  gcd(a,b)\overset{\color{#c00}{law}}=gcd(a−b,b)\overset{induct}{=}(a−b)x+by=ax+b(y−x).$$
This gcd $\rm\color{#c00}{law}$ is the reduction (descent) step employed in this algorithm to reduce the given gcd to a "smaller" one, where the size measure used on gcd argument pairs $\rm\,(a,b)\,$ is  their sum $\rm\,a+b.$
