Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in \[-1,1\]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open interval...) Someone hypothesised that the answer is yes, and further that every $x$ that is relatively irrational with $\pi $ has this property.


The question I am more interested in is the generalised version of this to arbitrary sequences. What are necessary and sufficient conditions for a sequence having subsequences converging to any point in the set of values the sequence visits? Does it have anything to do with properties like the function $f(n)$ being ergodic or mixing?


(suggestions for tags welcome in comments)



Existence of convergent subsequences for all values in range?
More conventional language: Are there values of $x$ such that the sequence $\sin(nx)$ is dense in the interval $[-1,1]$. The answer is yes, almost all $x$ have this property, in particular all $x$ such that $x/\pi$ is irrational. 


See Weyl's Criterion
<http://en.wikipedia.org/wiki/Weyl%27s_criterion>
for something (equidistributed) that implies much more than merely dense. And $nx$ mod 1 is equidistributed in $[0,1]$ if $x$ is irrational.


