Question: Is there a shift-invariant submultiplicative seminorm $||\cdot||$ of $\ell^\infty$ which satisfies the following property?

If $f:\mathbb{N}\rightarrow\mathbb{N}$ is an increasing function bounded below by some exponential function $cr^n$ for real numbers $c>0,r>1$, then the (bounded) sequence $||(\frac{f(0)}{f(1)},\frac{f(1)}{f(2)},\frac{f(2)}{f(3)},\dots)||<1$

(This, for example, excludes $\lim\sup$. If $f(n)=2^{\lfloor\frac{n}{2}\rfloor+1}$ then $f(n)>\sqrt{2}^n$, but the lim sup of the above sequence is $1$)

This comes up when estimating the asymptotic growth of certain "sparse" infinite binary trees. If there were such a norm, then the number of nodes at depth $n$ in such trees could not be bounded below by any such exponential function. By the way, I have no knowledge of functional analysis.