The answer is yes, and this question is strongly related to the
topic of [Hausdorff gaps](https://en.wikipedia.org/wiki/Hausdorff_gap).

You have specified a sequence of functions $x^{(k)}$ that is
decreasing modulo finite the usual order on
$\mathbb{R}^\omega/\text{Fin}$. That is, for each $k$, you have
$x^{(k+1)}_n< x^{(k)}_n$ for all but finitely many $n$.
Furthermore, if we let $z^{(k)}_n=k$ for all $n$ and $k$, we have
what is called an $(\omega,\omega)$-gap, because for any $k$ we
have $$z^{(k)}_n<z^{(k+1)}_n<\cdots<x^{(k+1)}_n<x^{(k)}_n$$ for
all sufficiently large $n$. Hausdorff proved that in this
situation, there is a function $n\mapsto y_n$ that fills the gap
(and you can find explicit constructions of the gap-filling
functions). So we have for each $k$ that $k<y_n<x^{(k+1)}_n$ for
all but finitely many $n$. And this ensures your desired
hypotheses.