I have the following question in my assignment which I couldn't solve:
Let $({a_n})$ and $({b_n})$ be two sequences, such that $\lim\limits_{n \to \infty}({a_n}{b_n}) =0 $
I have to prove if the following is true or false:
$\lim\limits_{n \to \infty}({a_n}) =0$ $or$ $\lim\limits_{n \to \infty}({b_n}) =0$
It seems true to me, for several reasons, but mainly because I couldn't find an example that contradicts the statement. I can not use limit arithmetic because I can't prove that the sequences are convergent, so every time I tried to do some calculation, I got stuck.
Your help is appreciated, thank you.