# Sequence of functions $f_n$ so that $\forall g \in C^0\left(\Bbb R,\Bbb R\right),\exists n \in \mathbb{N}, \cfrac{g}{f_n}$ is bounded

Context

I read this post which asks for an explicit norm on $C^0\left(\Bbb R,\Bbb R\right)$.

Obviously, usual norms don't work because of the fact $\Bbb R$ is infinite so using an integral would fail even on constant functions and $\sup$ would fail on affine functions.

The integral looking too difficult to fix for this space, I decided to try to fix the $\sup$ by crushing functions when they went to $\pm\infty$.

To do that, I assumed there was a sequence of functions $\left(f_n\right)_{n\in\mathbb{N}}\in \mathcal C^0\left(\Bbb R,\Bbb R\right)^n$ so that $\forall g \in C^0\left(\Bbb R,\Bbb R\right)^n,\exists n \in \Bbb{N}, \cfrac{g}{f_n}$ is bounded on $\Bbb{R}$. And I defined the norm to be $\|g\|=\sup \left|\cfrac{g}{f_n}\right|$ for the smallest such $n$.

Homogeneity and separation worked well but subadditivity did't work so I favored the post to have a look later to see if someone else answered and gave up.

And since then, I'm wondering whether such a sequence of functions exists.

The question

Is there a sequence of functions $\left(f_n\right)_{n\in\mathbb{N}}\in \mathcal C^0\left(\Bbb R,\Bbb R\right)^n$ so that $\forall g \in C^0\left(\Bbb R,\Bbb R\right)^n,\exists n \in \Bbb{N}, \cfrac{g}{f_n}$ is bounded on $\Bbb{R}$?

My first thought was to take something like $f_n=exp^n=\exp \circ \dots \circ \exp$ but I have no idea how to prove or disprove that it works... At sure I was fairly convinced it would work...

But then I thought of taking something like $f_n(x)= x^n$. Of course this doesn't work since if you took $g=\exp$, $\cfrac{g}{f_n}$ wouldn't be bounded for any $n$. But a few years ago, I would've thought it would work...

Then I tried to search for functions for which $f_n=\exp^n$ wouldn't work. And the only thing I could think of was Ackermann's idea of "creat[ing] an endless procession of arithmetic operations" (which I discovered while skim-reading this). The same way you go from $x\mapsto x+a$ to $x\mapsto ax$ by making the number of $a$s you add depend on $x$, we then have $x\mapsto a^x$ where the number of times you multiply by $a$ depends on $x$ so I guess we could apply the same kind of method once again that would make the $\exp^n$ fail as my $f_n$ but I can't conceive such functions. I'm not even sure they are possible to define...

$f_0$ defined by:

$f_0(0)=1$

$\forall n \in\Bbb{N}^*,f_0(n)=f_0(-n)=$ $n^{th}$ Ackermann number

And then the rest is just lines between those previously defined points.

And then $\forall n \in\Bbb{N}^*,f_{n+1}=f_n\circ f_0$

Let $g$ interpolate the values $g(n)=n \max_{k<n} |f_k(n)|$. Then $\left|\frac {g(x)}{f_n(x)}\right|\ge m$ for $x=m$, hence the quotient is not bounded.