Take for example $a_n=e^{in}\in\Bbb C$, or $b_n:=\sin n\in\Bbb R$.
We know by BW they admit converging subsequences; but are we able to find out some of them explicitly?
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I think you will find really interesting elements in this : https://math.stackexchange.com/questions/507823/subsequence-of-sin-n – Atmos Feb 28 '18 at 17:48
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As in writing down an explicit, formula-based sequence and its limit? – Feb 28 '18 at 17:49
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Probably this answer does not satisfy you, but if $\frac{p_1}{q_1},\frac{p_2}{q_2},\frac{p_3}{q_3},\ldots$ is the sequence of convergents of the continued fraction of $\pi$, then $\sin p_n\to 0$. These $p_n$s are $3, 22, 333, 355, 103993,\ldots$ – Jack D'Aurizio Feb 28 '18 at 17:50