We know that any sequence on $S^1$ must have a converging extracted subsequence, as $S^1$ is compact. Now, consider the sequence $a_n=(\cos(n),\sin(n))$. Could you find explicitly a subset of the natural numbers such that the corresponding subsequence converges? I don't even know whether it is possible to work it out, or whether there exists a nice representation of the solution. I thought about it and I found out a proof, but later I noticed that it actually contains an error. I'd prefer not to write here the proof, at least in this first time, since I could influence your reasoning. Thank you in advance,
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