Example that $\lim \sup (x_n\cdot y_n)<\lim \sup (x_n)\cdot \lim \sup (y_n)$ 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
$$\lim \sup (x_n\cdot y_n)&lt;\lim \sup (x_n)\cdot \lim \sup (y_n)$$

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.
Even values of x are zero, odd values are one.

y is the opposite.
Even values of x are zero, odd values are one.

y is the opposite.