I understand they're equal. I've proved it.
But why can it be the case that it is less than?
My number theory book says "$gcd(a,b) \le gcd(a, b-a)$, since the set over which we are taking the max for $gcd(a,b)$ is a subset of the set for $gcd(a, b-a)$".
I didn't quite understand this at all... Aren't they the same thing, therefore they should have the same set? Why is it that $gcd(a,b)$ is a subset of $gcd(a,b-a)$?