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I'm trying to determine all subsequential limit points of the following sequence

X_n = cos(n) 

Not sure how to decompose this into subsequences.

Anyone know how to tackle this problem?

Thanks!

1 Answers 1

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It is rather well known that the image of the integers under sine and cosine is dense in $[-1, 1]$. For a reference, see the related question here.

edit in response to op's comment

With the knowledge that $\cos(n)$ is dense in $[-1, 1]$, we shall show that any $x\in[-1,1]$ is a sub-sequential limit point. Indeed, given any $\epsilon > 0$ and some $n_0 > 0$, we can find $n > n_0$ such that $\cos(n) \in (x - \epsilon, x + \epsilon)$ so we can iteratively pick a sub-sequence that lies within this $\epsilon$-neighborhood. This shows that any point is a sub-sequential limit point of the sequence.

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    @Steven Stadnicki I've added the corrections. Thank you.2012-08-28