A problem on recurring points dividing [0, 1] into segments

[Kritixi Lithos]

Here is a problem I found somewhere else that I haven't managed to solve it yet.

Prove that for all real r and all natural n, the set of points {r×z mod 1 | z ∊ 0..n} ∪ {1} divide the interval [0, 1] into segments of at most 3 different lengths.