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Let my sequence be

$a_n=n\pi-\lfloor n\pi\rfloor$

This sequence is bounded in $[0,1)$ so if must have a convergent subsequence. In fact, it seems to me like it has infinitely many convergent subsequences, but I have no idea how I'd prove that.

I'm pretty sure it never hits the same point more than once but I'm also pretty sure that it doesn't hit every real number between $0$ and $1$. For example, due to the infinite nature of $\pi$'s decimals, I don't think it would ever hit numbers like $0.5$. Does it even hit any rational number?

If I was given a task to find all the limits of the convergent subsequences or at least give them some structure, how would I do it?

Luka Horvat
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    See this. (The upshot is the set of limit points is $[0,1)$.) – David Mitra Jul 15 '14 at 22:08
  • You can actually do much better than this. If you pick any integer polynomial with $0$ as a root and plug in $n\alpha$ it will always have this phenomenon modulo $1$. It's a very interesting study in Diophantine approximation. – Adam Hughes Jul 15 '14 at 22:48

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