The discrepancy of some real sequences

Let (lambda(n))(ngreater than or equal to0) be a sequence of real numbers such that there exists delta > 0 such that \lambda(n+1) - lambda(n)\ greater than or equal to delta, n = 0, 1,.... For a real number y let (y) denote its fractional part. Also, for the real number x let D(N, x) denote the discrepancy of the numbers {lambda(0)x},..., [lambda(N-1)x}. We show that given epsilon > 0, D(N, x) = o (N-1 (2) over bar(logN)(3 (2) over bar+ epsilon)) almost everywhere with respect to Lebesgue measure.