Abstract. The probability inequality for max k≤n S k , where S k = k j=1 X j , is proved under the assumption that the sequence S k , k = 1, . . . , n is a supermartingale. This inequality is stated in terms of probabilities P(X j > y) and conditional variances of random variables X j , j = 1, . . . , n. As a simple consequence the well-known moment inequality due to Burkholder is deduced. Numerical bounds are given for constants in Burkholder's inequality.