# If $a_n$ doesn't have any subsequence that converges, can $|a_n|$ converge?

If a sequence $$a_n$$, $$n\in\mathbb{N}$$ doesn't have any convergent subsequence can $$|a_n|\rightarrow a$$, $$a\in[0,\infty)$$?

My intuition says that this isn't possible but I'm not sure how to prove it..

• Well, if the absolute sequence converges then so does the sequence of positive terms and the sequence of negative terms (one of which, at least, must be infinite). – lulu Jan 21 at 20:32
• Hint: do you know any conditions which imply that $a_n$ does have a convergent subsequence? Your sequence must violate those conditions. – Wojowu Jan 21 at 20:32

If $$(|a_n|)$$ is convergent, then $$(a_n)$$ is bounded. Now invoke Bolzano-Weierstraß. Conclusion?