$\DeclareMathOperator*{\plim}{plim}$ Say we have two sequences of random variables $\{Z_{n,j}\}, \{X_{n,j}\}$, indexed by both $n$ and $j$.
Also assume that $\lambda_j^Z$ and $\lambda_j^X$ are non-random and for each $j$:
$\plim_{n\to \infty} Z_{n,j} = \lambda_j^Z$
$\plim_{n\to \infty} X_{n,j} = \lambda_j^X$
Questions about whether the following statements are valid:
(1) $\plim_{n\to \infty} \frac{Z_{n,j}}{X_{n,j}} = \frac{\lambda_j^Z}{\lambda_j^X}$
(2) If $\lim_{j\rightarrow\infty} \frac{\lambda_j^Z}{\lambda_j^X} = 0,$ then $\plim_{j\rightarrow \infty}\big[\plim_{n\to \infty} \frac{Z_{n,j}}{X_{n,j}}\big] = 0$
The first should be a consequence of Slutksy's theorem, and the second also seems correct, though I feel like I'm missing something.