Suppose I have the following condition:
$$\max \left\{ \frac{f(x)}{h(x)},\frac{g(x)}{h(x)} \right\}, \forall x \in \mathbb{R}$$ where $f(x),g(x),h(x)$ are all non-negative (real) functions. Is it theoretically correct to state that this is equivalent to: $$\frac{\max \left\{ f(x),g(x)\right\}}{h(x)}, \forall x \in \mathbb{R}$$ I'm having some trouble proving this. Your help is much appreciated! Thanks!