I am trying to find a non-constructive proof for a simple example problem:
Let set $B=\{g||\forall x, g(x)>0,g'(x)>0\}$;
$A=\{f||f=v\cdot h, \text{and } v,h\in B\}$. Prove that $A=B$.
I can first prove that $f'=v'h+vh'>0$,
So any $f\in B$;
Therefore $A\subseteq B$.
But is it possible to find a general way (i.e. not by construction) to prove that $B\subseteq A$?
My try: suppose that $g=h\cdot (g/h)$. We want to show that there exists $h$ such that $h'>0$ and $(g/h)'>0$.
$(g/h)'=(g'h-gh')/h^2$
It is greater than zero if and only if $g'h-gh'>0$;
That is, $h'<\frac{g'h}{g}$.
Where I can find a book/paper on the methods of proving these things? I guess for differentiable functions, we can prove that both sets are the solution set of the same differential equation?