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

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 ?

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.
• this is cool, but if $u_n$ and $v_n$ were always strictly positive ? – Noctisdark Nov 20 '17 at 1:22
• Then you can proceed like above and swap 0 in both cases with a sequence that converges to 0, like $1/n$. – Cornman Nov 20 '17 at 1:25
• Then put $u_{n}=1/n$ for $n$ even and $u_{n}=1$ for $n$ odd, $v_{n}=1$ for $n$ even, $v_{n}=1/n$ for $n$ odd, then $u_{n}v_{n}=1/n$. – user284331 Nov 20 '17 at 1:27