If we are given monotonic real sequences $(a_n)_{n\ge1}, (b_n)_{n\ge1}$ then does the limit $\lim_{n \to \infty} a_n b_n $ always exists (+-infinity is also considered as limit point)? The case that needs looking into is obviously the one when, for example $a_n \to \infty$ and $b_n \to 0$ .
I've been thinking a bit and i can't seem to find a counterexample (i am pretty sure that a limit does not always exist)
Thanks in advance!