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.



Sequence of tending to zero functions that majorizes any other tending to zero function
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 $g+h \notin E\_{K,n}$


