This is a short question, I already managed to prove using definitions that $$\lim \sup (x_n\cdot y_n)\le \lim \sup (x_n)\cdot \lim \sup (y_n)$$
But I'm having trouble coming up with an example such that $$\lim \sup (x_n\cdot y_n)<\lim \sup (x_n)\cdot \lim \sup (y_n)$$
I tried to consider alternative sequences but I'm not sure if I'm doing it right. I'm considering the following right now. $$x_n=(1,0,1,0,...)$$ $$y_n=(0,1,0,1,...)$$ $$x_n\cdot y_n=(0,0,0,0,...)$$ $\lim \sup x_n \cdot y_n=0$ as there sequence is convergent. But $\lim \sup x_n = 1$ and $\lim \sup y_n =1$ So it appears the inequality holds. I just need a confirmation that what I'm doing is right. Sorry if this is a redundant question, I'm just learning this concept so it's a little fuzzy for me.
Note that $(x_n)$ and $(y_n)$ are non-negative.