Convergence of product of random variables

Question

If random variables $X_n\rightarrow X$ in distribution and $Y_n\rightarrow c$ in probability, it follows from Slutsky’s theorem that $X_nY_n\rightarrow cX$ in distribution. If we further assume that $Y_n\rightarrow c$ almost surely and $c=0$, is the following true: $$X_nY_n\rightarrow 0,\ a.s.?$$ Thanks!

Answer 1

Arrange the intervals $(\frac {i-1} {2^{n}}, \frac i {2^{n}})$ in a sequence, say $I_1,I_2,...$. Let $X_n=n$ on $I_n$ and $0$ elsewhere. Let $Y_n =\frac 1 n$. Then $X_n \to 0$ in probabilty (hence in distribution), $Y_n \to 0$ almost surely but $X_nY_n$ does not $\to 0$ almost surely. [ In fact, for every $\omega$, $X_n(\omega)Y_n(\omega)=1$ for infinitely many $n$.]