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  1. Determine at least three limit points for the set {$\sin(n)$: n a positive integer}.

  2. How many limit points does the set {$\sin(n)$: n a positive integer} have?

Our professor gave us a definition to use for limit point in order to differentiate between a cluster point. The definition is as follows:

Let $S$ be a nonempty set of $\mathbb{R}$ where $S \subseteq \mathbb{R}$

Let $x \in \mathbb{R}$

We say that $x$ is a limit pont of $S$ if:

For each $\epsilon > 0 $ there is an element of $S$ in $(x-\epsilon$, $x+\epsilon)$.


With our first question and the given definition, wouldn't the numbers $\sin(1)$, $\sin(2)$, $\sin(3)$ work?

Choose the $x = \sin(1)$

We then have:

$\sin(1)-\epsilon < \sin(1) < \sin(1) + \epsilon$

Which seems trivially true. The same argument would follow for the next two points. However, the next part is where myself and a few of my other classmates are completely lost.

A few questions for 2.
What direction should I head for the second half? The professor mentioned Kronecker's theorem. If possible, could someone give a breakdown of Kronecker's theorem and it's applications?

Kronecker's Theorem: http://mathworld.wolfram.com/KroneckersApproximationTheorem.html

How can I find the cardinality of the set of limit points?

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    You might be able to work out the answers after having a look at http://math.stackexchange.com/questions/63526/showing-sup-sin-n-n-in-mathbb-n-12012-02-10
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    I adjusted my definition. I incorrectly transposed it from my notes.2012-02-10
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    Consider irrational rotation on the circle. That is, consider $R_r: S^1\rightarrow S^1$, $R_r(x)= (r + x)(\mathrm{mod} 1)$, where $S^1$ is the unit circle given by $[0,1]/\sim$, where ~ is the equivalence relation identifying $0$ with $1$ (you simply wrap the interval $[0, 1]$ into a circle, gluing the ends). Now for $r$ try some nice irrational values, like $\pi/2$, $\pi/4$, $\pi/6$, and start applying $R_r$ repetitively to zero: $R_r(0)$, then $R_r(R_r(0))$, and so on... (hint: $\{x_n\}_{n\in\mathbb{N}}$ is dense in $[0,1]$, where $x_n = R_r\circ\cdots\circ R_r(0)$ $n$-times.2012-02-10
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    I was just reading this en.wikipedia.org/wiki/Irrational_rotation2012-02-10
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    @arete: Well, think about it. If it still doesn't make sense, comment, and I'll make my comment into an answer.2012-02-10
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    We're currently attempting Kronecker's route, but I'm interested in your approach.2012-02-10
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    "$\pi$ is irrational" is crucial in here.2013-04-07

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