Does there exist a sequence of decreasing continuous functions $(f_n)_{n\in\mathbb{N}}$ satisfying the following two conditions?

- For every $n\in\mathbb{N}$, $\lim_{x\to\infty}f_n(x)=0$;
- For any other decreasing continuous function $g$ tending to zero at infinity, there exists $n\in\mathbb{N}$ so that $\frac{g}{f_n}$ is a decreasing to zero function.

The spirit is quite clear, it is tempting to say that we can approach the constant function equal to zero with a sequence that eventually majorizes every other function tending to zero at infinity. But, I cannot come up with such an example satisfying the second condition above and I am not even sure that this is true. Of course, any idea allowing to answer this with restrictions (for all $g$ convex or satisfying some regularity conditions,... or asking only that $\lim_{x\to\infty}\frac{g(x)}{f_n(x)}=0$ or $\frac{g}{f_n}$ just decreasing,...) is welcome.