This was first posted to SE, but now I think its better to be posted here.
For what positive real numbers $\alpha$, the sequence $a_n = \frac{\lfloor n\alpha\rfloor}n $ is (not necessary strictly) increasing for sufficiently large indexes ? ($\lfloor x\rfloor$ is the integer part of $x$).
Only if $\alpha$ is an integer (in which case the sequence is constant). Suppose $\alpha$ is not an integer. By subtracting $\lfloor \alpha \rfloor$, we may assume $0 < \alpha < 1$. Then there are arbitrarily large $n$, such that $0 < \lfloor n\alpha \rfloor = \lfloor (n+1)\alpha \rfloor$, so $$ \frac{\lfloor n\alpha \rfloor}{n} = \frac{\lfloor (n+1)\alpha \rfloor}{n} > \frac{\lfloor (n+1)\alpha \rfloor}{n+1} .$$
