if we have a sequence of random variables $X_n\to 0$ a.s. (but we have no idea how fast), can we always build a sequence $a_n\to\infty$ such that $a_nX_n\to 0$ still? If not what is a counterexample.
I tried using $a_n:=\frac{1}{\sup\{\omega: X_n(\omega)^{2}\}}$ but this does not work
How about if $X_n \sim \frac 1{\log n} \to 0$ But $nX_n \sim \frac n{\log n} \to \infty$. Presumably $a_n \ge n$ because you are picking them.