$|a_n|$ is diverging.
How do you prove $a_n$ (without absolute value) has a subsequence converging to a finite limit?
I know that if a sequence has two subsequences converging to different numbers, then the sequence is diverging. May I use the opposite lemma?
The sequence $|a_n|$ doesn't diverging to $\infty$. So you may say the sequence has no limit.