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?
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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