On a criterion of uniform distribution

We give an extension of a criterion of van der Corput on uniform distribution of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions ${\mathrm{\Delta }}^2{x}_n\to 0,n^2{\mathrm{\Delta }}^2{x}_n\to \mathrm{\infty }$. Our proof is straightforward and uses a Diophantine approximation by rational numbers, while van der Corput’s approach is based on some estimates of exponential sums.