I was reading an Advanced Calculus textbook and came up with the following question: From Bolzano-Weierstrass Theorem, I know there is at least one convergent subsequence of $x_{n}=sin(n^3+n+1)$. But How many are there? Is the image of $x_{n}$ dense in [-1,1]? What tools can I use to investigate?
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The set $\left\{n \pmod \pi:n\geq 0\right\}$ is dense in $(0,\pi)$ because if it wasn't then $\pi$ would be rational. It seems like $\left\{n^3+n+1 \pmod \pi:n\geq 0\right\}$ should be dense in $(0,\pi)$ too which would imply that the image of $x_n$ is dense in $[-1,1]$. And then that implies that there's infinitely many convergent subsequences. Proof of the first assertion can be found here: Positive integer multiples of an irrational mod 1 are dense
