Discussion
The sequence \(a_{n} = (-1)^{n} + \dfrac{6}{n^{2}}\) has :
Answer: C
The sequence has two limit points, 1 and -1.
The term \(6/n^{2}\) tends to 0, so the behaviour of \(a_{n}\) is decided by \((-1)^{n}\). Along even \(n\), \(a_{n} = 1 + 6/n^{2} \to 1\); along odd \(n\), \(a_{n} = -1 + 6/n^{2} \to -1\). Every subsequence eventually lies near one of these two values, so they are the only limit points.
'No limit point' is impossible for a bounded sequence (Bolzano-Weierstrass). 'One limit point' would need the sequence to converge, which it does not, since its even and odd terms approach different values. 'More than two' would need a third cluster value, and there is none.
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