I am wondering how the sandwich formula of the variance is derived.
Assume $A_n$ and $X_n$ are two random variables in $\mathbb{R}$ such that $A_n$ converges in probability to a constant $A_0$ and $\lim_{n\to\infty} V(X_n) = B_0$. How to show that $\lim V(A_n X_n) = A_0^2B_0$? This is generally stated in many textbooks. I am wondering if we do not need more conditions. Something like the $\nu$-th moment of $\lvert A_n X_n \rvert$ is bounded, for some $\nu > 2$. How to show the formula of the asymptotic variance of $A_n X_n$?