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I have to determine whether the following statement is true or not:

Let $(a_n)$ and $(b_n)$ be two sequences such that $$\lim_{n\to\infty}(a_nb_n)=0$$ then either $\lim_{n\to\infty}(a_n)=0$ or $\lim_{n\to\infty}(b_n)=0$

It seems like the statement is true and I can't find any counterexample for it. But for some reason I fail to prove it (the standard product rule doesn't work here because we know nothing about the convergence of $(a_n)$ and $(b_n)$). I'd appreciate any ideas on how to prove it.

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  • $\begingroup$ @Dr.MV - looks like the statement is false (see Timbuc example). $\endgroup$ – user228971 Apr 5 '15 at 19:08
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    $\begingroup$ If you add the requirement that both limits $\lim a_n$ and $\lim b_n$ have to exist then the statement becomes true. $\endgroup$ – Winther Apr 5 '15 at 19:53
  • $\begingroup$ @Winther - What if we know that $\lim_{n\to\infty}(b_n)=1$? Can we say that $\lim_{n\to\infty}(a_n)=0$? $\endgroup$ – user228971 Apr 5 '15 at 20:01
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    $\begingroup$ Yes since $a_n = \frac{a_nb_n}{b_n}$ so $\lim a_n = \frac{\lim a_nb_n}{\lim b_n} = \frac{0}{1} = 0 $ $\endgroup$ – Winther Apr 5 '15 at 20:11
  • $\begingroup$ @Winther - brilliant, sir. Thank you. $\endgroup$ – user228971 Apr 5 '15 at 20:13
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Hint:

Take

$$\{a_n\}=\{0,1,0,1,0,\ldots\}\;\;,\;\;\;\;\{b_n\}=\{1,0,1,0,1\,\ldots\}$$

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  • $\begingroup$ Oh, gosh, you made me feel so stupid. Thank you! $\endgroup$ – user228971 Apr 5 '15 at 19:06
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    $\begingroup$ $\{\}$ is used for sets, so you're saying $\{a_n\}=\{0,1\}$. Overall this seems like abuse of notation. Maybe it should be denoted with $(a_i)_{i\in\mathbb Z^+}=(0,1,0,1,\ldots)$. $\endgroup$ – user26486 Apr 5 '15 at 22:57
  • $\begingroup$ @user31415 Many authors agree with the notation I use: $\;\{a_n\}\;$ meaning: the sequence (which is a set) with elements $\;\{a_1,a_2,..,a_n,...\}\;$, and in $\;\{a_n\}=\{0,1,0,1,\ldots\}\;$ the meaning is the elements are numbered. You can check the books by Purcell-Varberg, Thomas, Thomas-Finney, Stewart, Swokowski, Spivak, etc. The meaning is usually pretty clear from the context. $\endgroup$ – Timbuc Apr 6 '15 at 8:21
  • $\begingroup$ What if for all but a finite number of terms $b_n\geq c, c>0$? Can we say then that $\lim_{n\to\infty}(a_n)=0$? (see my reasoning in the comments above) $\endgroup$ – trfv Apr 6 '15 at 10:36
  • $\begingroup$ @trfv Yes, I think that's correct. $\endgroup$ – Timbuc Apr 6 '15 at 14:35

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