Let us consider a countable set of sequences of positive numbers $\{(x_n^{(1)}),(x_n^{(2)}),\dots\}$ for which we have

• $(\forall k\in\mathbb{N}) \ \lim_n x_n^{(k)} = +\infty$, and
• $(\forall k\in\mathbb{N}) \ \lim_n\frac{x_n^{(k+1)}}{x_n^{(k)}}=0$ (the growth rate of the sequences is in the descending order).

My question is the existence of a minimal element, i.e. is there a sequence of positive numbers $(y_n)$ such that $\lim_n y_n=+\infty$ and for every $k\in\mathbb{N}$ we have $\lim_n\frac{y_n}{x_n^{(k)}}=0$?

Of course, any bounded sequence satisfies the second condition, but that is not satisfactory to me.

An example of the above situation where the existence of $(y_n)$ is obvious is $x_n^{(k)}=n^{\frac{1}{k}}$, where we can take $y_n=\ln n$.