If $\lim_{n\to \infty}a_n\cdot b_n=0$ then either $\lim_{n\to \infty}a_n=0$ or $\lim_{n\to \infty}b_n=0$ or both. [closed]

Prove or disprove: If $\lim_{n\to \infty}a_n\cdot b_n=0$ then either $\lim_{n\to \infty}a_n=0$ or $\lim_{n\to \infty}b_n=0$ or both.

The answers say it is not necessarily true. I can't find a counter example, however. Would appreciate your reply.

• Do limits of $a_n,b_n$ exist? Commented Feb 24, 2015 at 13:48

Take $a_n = 0$ if $n$ is odd and $a_n=1$ if $n$ is even and take $b_n = 1$ if $n$ is odd and $b_n=0$ if $n$ is even.
Let $a_n=0$ if $n$ is odd and $1$ if $n$ is even and $b_n=1$ if $n$ is odd and $0$ if $n$ is even.
Proof: Without loss of generality, let $$a_n$$ have a limit. If the limit is zero, it is done. If the limit is nonzero, without loss of generality assume it is a positive value $$m$$. From some point onwards, $$a_n$$ would have only positive terms. Assume that $$b_n$$ doesn't have limit $$0$$, so it must have infinitely many terms at least some distance (call it $$k$$) from $$0$$. For arbitrarily large $$n$$, the product would have absolute value at least $$km - \varepsilon$$ for $$\varepsilon$$ tending to $$0$$. Therefore, $$b_n$$ must have limit $$0$$.