# Convergent Subsequence in $\mathbb R$

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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Is it fair to say that since those subsequences are the only ones I have to consider because they are the only two convergent subsequences? Other than ones like $\langle a_{4k} \rangle$ which are contained within the one considered? – Moderat Aug 14 '12 at 21:41
You can take a subsequence that includes infinitely many terms from each of those subsequences and it will still converge to 0. Like I said, $\textbf{every}$ subsequence will converge to 0, not just ones that have only even terms or only odd terms. – Francis Adams Aug 14 '12 at 21:45
Never mind, of course you are right, because the denominator $\to \infty$ so the sequence $\to 0$, sorry I lost perspective. Thanks again! – Moderat Aug 14 '12 at 21:51