Sequence of tending to zero functions that majorizes any other tending to zero function Does there exist a sequence of decreasing continuous functions $(f_n)_{n\in\mathbb{N}}$ satisfying the following two conditions?

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*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.
 A: Let $C_d$ be the space of all decreasing continuous functions tending to 0 at infinity, equipped with the sup metric $d_\infty$.  Note this is a complete metric space.
Suppose such a sequence $f_n$ did exist.  Then for every $g \in C_d$ there would exist $n$ such that $g/f_n$ is decreasing to 0; since $g/f_n$ is continuous, it would therefore be bounded, so there would exist $K$ such that $|g| \le K |f_n|$. So if we let $E_{K,n} = \{g \in C_d : |g| \le K |f_n|\}$, we would have $\bigcup_{k,N=0}^\infty E_{K,n} = C_d$.  I claim that each $E_{K,n}$ is nowhere dense, and so this contradicts the Baire category theorem.
It is easy to see that $E_{K,n}$ is closed in $C_d$.  To see it also has empty interior, fix $g \in E_{k,n}$ and $\epsilon > 0$.  Since $K f_n$ vanishes at infinity, there is $M$ so large that $K |f_n(x)| < \epsilon/3$ for all $x > M$.  Choose $h \in C_d$ with $\sup |h| < \epsilon$ and with $h(x_0) > 2 \epsilon/3$ for some $x_0 > M$.  Then $g+h \in C_d$ with $d_{\infty}(g, g+h) < \epsilon$, but $(g+h)(x_0) > \epsilon/3$ so $g+h \notin E_{K,n}$
