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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 ?

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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.

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

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