Suppose I have a sequence of random variables $X_{nm}$ such that I know that $X_{nm} \xrightarrow{D} X_n$ as $m\rightarrow \infty$ at each fixed $n$, and for the sequence $X_n$. I also know that $X_n \xrightarrow{D} X_0$ as $n\rightarrow\infty$.
Is it possible for me to say that there exists a sufficiently slowly increasing sequence $N = n(m)$ such that $X_{Nm} \xrightarrow{D} X_0$ as $m\rightarrow \infty$?