I have recently been trying to solve a problem involving infinite series, and there is one which I have been able to reduce to a problem in Number Theory, which, however, still gives me trouble. The question is as follows:
$\forall \alpha \in (0;1)$, prove that the sequence $x_n = ||n\alpha||$ can never tend to $0$ as $n \to \infty$ (here I define $||x|| := \min{(\{x\}; 1-\{x\})} \, \forall x\in\mathbb{R}$ as the distance from $x$ to its nearest integer).
The proof is very much straightforward for the case when $\alpha \in \mathbb{Q}$, but the case $\alpha \in \mathbb{R}\setminus\mathbb{Q}$ has eluded me for some time.
I have figured that if I could prove that for any monotonic increasing (or decreasing) sequence $r_n$ of rational numbers (all in the interval $(0;1)$) such that $\lim_{n\to\infty}r_n = \alpha \in (0;1)$, the sequence $||nr_n||$ cannot tend to $0$, the irrational $\alpha$ case would follow (and the rational, too, for that matter).
With this, however, I have also had difficulty proving. I should really appreciate any advice you can give me, but bear in mind that I do not want complete solutions, or even significant portions of such. I need only a hint. Thank you.