Say that $\{X_n\}$ are none-negative random variables converging almost surely to zero, and $Y_n$ are a sequence of i.i.d random variable, non-negative, with finite $r$-moment for any $r>0$, and whose distribution does not depend on $n$. Then $X_nY_n\rightarrow 0$ in probability (using Markov inequality), but is it true that $X_nY_n\rightarrow 0$ almost surely?
If $Y_n\rightarrow Y$ almost surely then it was true due to the continuous mapping. But is it true when $Y_n$ has no limit (but has finite moments that do not depend on $n$)?
Edit: answer given in the comments
Julius's comment answers my question. If the moments are not bounded (that is, there is a constant $C$ such that $EY_n^r<C$ for all $r>0$, then we can't deduce almost sure convergence.