Convergence in probability of a sum of dependent random variables to 0

Question

Suppose we know that $A_n + B_n \xrightarrow{p} 0$ as $n \rightarrow \infty$, and we know that $A_n \xrightarrow{d} N(0,1)$. Can we say that $B_n \xrightarrow{d} N(0,1)$? Note that $A_n$ and $B_n$ are definitely not independent.

Answer 1

We have that $$ B_n=A_n+B_n-A_n. $$ $A_n+B_n\to0$ in probability as $n\to\infty$ and $A_n\to Z$ in distribution as $n\to\infty$, where $Z$ is a standard normal random variable. By Slutsky's theorem, $A_n+B_n-A_n$ converges to $0-Z=-Z$ in distribution as $n\to\infty$. Hence, $B_n\to Z$ in distribution as $n\to\infty$.