Problem: Write all the cluster points of the set
$$ A = (-1)^n: \quad\forall n\in\mathbb N $$
I'm puzzled. I don't know how I can find them aren't there too many?
Any help is appreciated!
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1$\begingroup$ What is the definition of cluster point you have in mind? $\endgroup$– BcpicaoApr 25, 2020 at 23:40
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$\begingroup$ A point $x∈R$ is called a cluster point of set A subset of R if every neighborhood V of $x$ contains an element a∈A different from $x$ $\endgroup$– raydiiiiApr 25, 2020 at 23:42
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$\begingroup$ @Bcpicao is it only 1???? $\endgroup$– raydiiiiApr 25, 2020 at 23:43
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1$\begingroup$ I will add this as a comment as maybe I am just unfamiliar. I would think $A=\{-1,1\}$ and so there are no cluster points in this set. What the answer below to me is describing is a convergent subsequence. The definition of cluster point requires that there be a point different from $x$ from the set within every arbitrary neighborhood of $x$. Take the neighborhood with radius $\varepsilon=1/2$ about $x=1$, there are no other points from $A$ different from $1$ within that neighborhood. $\endgroup$– user43138Apr 26, 2020 at 0:01
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$\begingroup$ @Frudrururu, exactly! $\endgroup$– KoroApr 26, 2020 at 0:01
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1 Answer
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The sequence $\left( -1\right)^n$ alternates between $-1$ and $+1$. Every odd $n$ gives $-1$ and every even $n$ gives $+1$. So there are infinitely many $n$ such that the sequence is equal to either of these values, so they are both cluster points. Conversely, these are the only two discrete values of the sequence so there are no other values that could possibly be cluster points other than $\pm 1$.
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