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Let $\langle a_n \rangle = \dfrac{(-1)^n}{1+n}$ be a sequence in $\mathbb R$.

Considering the limit point(s) of this sequence and the subsequences that converge to this point, I have two subsequences: $ a_{2k} = \frac{1}{1+2k} \to 0 \text{ and } a_{2k+1} = \frac{-1}{1+2k+1} \to 0 $

If the question asks for the limit points of the sequence, and a subsequence that converges to this limit point, do I leave out the second subsequence? I don't think I've missed a limit point, but it doesn't hurt to check.

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If you want to find the limit points, you have to think about the subsequential limits for every subsequence, not just those two. But luckily, the original sequence converges (to 0) and in a convergent sequence, every subsequence converges to the same limit as the whole sequence does.

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    Never mind, of course you are right, because the denominator $\to \infty$ so the sequence $\to 0$, sorry I lost perspective. Thanks again!2012-08-14