Assume that we have two sequences of processes $X_{n} \in \ell_{1}$ and $Y_{n} \in \ell_{1}$ with non-negative components. Next, we know that $X_{n,i} \leq Y_{n,i}$ for all $i \in\mathbb{Z}_{+}$ and both $X_{n}$ and $Y_{n}$ converge to $a \in \ell_{1}$ in the norm of $\ell_{1}$. Next, assume that for some integers $q$ and $k$ we know that $n^q||Y_{n} - a||_{k}$ is bounded.
What can we conclude about $$ n^q||\frac{n}{n-1}X_{n} - a||_{k}? $$
Is it bounded?
No. Take $X_n=Y_n=a=e_1=(1,0,\dots)$. Then the assumption is true for all $q>0$ and $k$, but $ \frac n{n-1}X_n-e_1 = \frac1{n-1}e_1, $ so $n^q \| \frac n{n-1}X_n-e_1\|_k$ is bounded only if $q\le1$.
(I supposed $\|\cdot\|_k$ means the $l^k$-norm.)