Show that for each irrational number $x$ the set of limit points of the sequence
$(a_n)_{n\in\mathbb{N}}=nx-[nx]$
is the interval $[0,1]$.
($[x]$ is the largest integer $\leq x$)
Any ideas how to prove that?
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There is another proof, using the pigeonhole principle, here. – Brian M. Scott Nov 22 '15 at 22:41