Zero times an expression that may not have a probability limit Suppose we consider a product $a_nb_n$ in which for each $n\geq 1$, $a_n$ and $b_n$ are both random variables. Suppose we know that $a_n$ converges in probability to $0$ and $b_n$ does not necessarily converge in distribution to a random variable. Can we then say $a_nb_n$ converges in distribution to $0$?

No. Let $a_n = \frac 1n$, and $b_n = n$ (constant random variables). Then $a_n \to 0$ a. s. (and hence in probiability), and $a_n b_n \to 1 \ne 0$ a. s. (and hence in distribution). As the constant 1 and the constant 0 have different distribution, we cannot have $a_n b_n \to 0$ in distribution.
