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