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..
2026-03-25 15:41:14.1774453274
If $a_n$ doesn't have any subsequence that converges, can $|a_n|$ converge?
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If $(|a_n|) $ is convergent, then $(a_n) $ is bounded. Now invoke Bolzano-Weierstraß. Conclusion?