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 xₙ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions Δ²xₙ→ 0, n²Δ²xₙ→ ∞. 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.