Can please someone help me with the following problem.

Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational.

Now I need to solve the inequality $nx \; \mathrm{mod} \; 1 < v$ with respect to $n$, for some given small $v$.

According to Equidistribution Theorem, this sequence is uniformly distributed on (0,1). And thus we definitely know that there is an infinite number of those $n$'s. But what can we say about these solutions themselves? In particular, I need to know if the distance between two consequtive solutions is limited or not.

It seems to be true to me, but maybe I'm missing something... and can't figure out a hard proof.