# True or false statement about a simple limit of product

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.

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

$$\{a_n\}=\{0,1,0,1,0,\ldots\}\;\;,\;\;\;\;\{b_n\}=\{1,0,1,0,1\,\ldots\}$$
• $\{\}$ 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 $(a_i)_{i\in\mathbb Z^+}=(0,1,0,1,\ldots)$. – user26486 Apr 5 '15 at 22:57
• @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. – Timbuc Apr 6 '15 at 8:21
• 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) – trfv Apr 6 '15 at 10:36