Product of limits: If $\lim_{n\to\infty} u_nv_n=0$, does it mean that $\lim_{n\to\infty} u_n=0$ or $\lim_{n\to\infty} v_n=0$?

Question

I was doing a homework and while I was trying different solutions, I found this instance that puzzled me.

If $\lim_{n \rightarrow +\infty} u_n * v_n = 0$ is it true that:

$\lim_{n \rightarrow +\infty} u_n = 0$ or $\lim_{n \rightarrow +\infty} v_n = 0$.

Can I get a counterexample or a proof please ?

Answer 1

No, for $u_{n}=0$ for $n$ even and $u_{n}=1$ for $n$ odd, $v_{n}=1$ for $n$ even, $v_{n}=0$ for $n$ odd, then $u_{n}v_{n}=0$ but neither $\{u_{n}\}$ nor $\{v_{n}\}$ converge.