I'm having trouble understanding the steps in showing that a Homothetic production function's cost function must be expressible in the form $C(w, q) = a(w)b(q)$.
Since the production function is homothetic, I know that the optimal cost minimizing input ratio given the same input costs must be exactly the same for different quantities of output i.e.
$$ \frac{H^j(w,q)}{H^i(w,q)} = \frac{H^j(w,q')}{H^i(w,q')} $$
Rearranging this we arrive at the ratio:
$$ \frac{H^i(w,q)}{H^i(w,q')} = \frac{H^j(w,q)}{H^j(w,q')} $$
This makes sense because in order to the maintain the input ratio constant, we would need both cost-minimizing inputs to increase by the same factor when going from $q$ to $q'$. However, the proof then states that "for the above to true it is clear that the ratio must be independent of w, thus, setting $q' = 1$"
$$ \frac{H^1(w,q)}{H^1(w,1)}= \frac{H^2(w,q)}{H^2(w,1)} = \cdots =\frac{H^m(w,q)}{H^m(w,1)} = b(q) $$
and so
$$ H^i(w,q) = b(q)H^i(w, 1) $$
The steps after this to get to $C(w, q) = a(w)b(q)$ is pretty understandable and straightforward for me.
However, what I don't understand is how you can clearly see that the ratio must be independent of w. Surely the ratio just states that for a given w the inputs must increase by the same factor/ratio. But how does that imply that the ratio has to be exactly the same for all w?
Edit: for those who are asking, this comes from an exercise question in Frank Cowell's textbook: Microeconomics: Principles and Analysis.