# Cluster point of $a_{n}:=n+(-1)^{n}n$

I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution?

The subsequence diverges for increasing even $n$ since $2n$ grows infinitely.
The subsequence converges to $0$ for increasing odd $n$ since $n+(-1)n=0$

So my conclusion is that there is only one cluster point and that is $0$.

Am I fine with this?

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"The" cluster point? Why do you think it has one, and only, one such a point? Anyway, what you did is correct. –  DonAntonio Dec 22 '12 at 12:59
how can i find more then ? –  doniyor Dec 22 '12 at 13:00
There are no more, but perhaps you could explain why your proof does actually show that zero is the only cluster point of that sequence...Check that you did so partioning the set of natural numbers in two disjoint subsets: even and odd natural numbers. This is enough and necessary to find all possible cluster point, and thus you should, perhaps, either prove it or at least mention it. –  DonAntonio Dec 22 '12 at 13:02
oh okay, thanks –  doniyor Dec 22 '12 at 13:05
As an aside which may or may not help, if you were computing in the extended reals rather than the reals, $+\infty$ would also be a cluster point. –  Hurkyl Dec 22 '12 at 14:15
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