If $\limsup x_n = x$, $\lim y_n = y$, $x_n, y_n > 0$, then does $\limsup (x_n y_n)= xy$?

Question

I have to prove the following statement, but I can't. If $\limsup x_{ n }=\, x,\lim y_{ n }=\, y, \, x_{ n },y_{ n }>0$, then $\limsup (x_{n}y_{n})=xy$.

Will you give me some hint or solution?

Answer 1

let $s_n = \sup_{k \ge n} x_n$. We have $s_n \uparrow x$. By continuity we have $s_n y_n \to xy$.

Find an upper bound for $\limsup_n x_n y_n$:

Clearly $x_n y_n \le s_n y_n$, hence $\limsup_n x_n y_n \le xy$.

Furthermore, we have a subsequence $x_{n_k} \to x$, and again, we have $x_{n_k}y_{n_k} \to xy$.

Find a lower bound for $\limsup_n x_n y_n$:

Since we have $\lim_n x_{n_k}y_{n_k} = xy$, we have $\limsup_n x_n y_n \ge xy$.

Answer 2

Hint. If $\limsup x_n =x$ then you can find a subsequence $\langle x_{n_k}\rangle$ such that $\lim x_{n_k}=x$. In general, if $a$ is a limit point of $\langle x_n\rangle$ then you can find a subsequence of $\langle x_n\rangle$ which converges to $a$.