For any ε > 0, because R(x) has supremum 1/2, so we might as well set ε.

According to the definition of Riemannian function, r (x) >; = ε, x must be a rational number, x=q/p, q, p is a positive integer, and q/p is a reduced true fraction.

r(q/p)= 1/p & gt； =ε

2 & lt= p & lt= 1/ε

Namely 2

Therefore, the value of p is limited at most.

Because 1

To sum up, q/p is limited at most, that is, r (x) >; = ε has only a finite number of solutions.