This note applies the method of indicator functions to derive a general inequality for the probability that at least one out of n events occurs. The result is in the spirit of the early work of [Chung, 1941,1943, Ann. Math. Statist. 12, 328–338; Ann. Math. Statist. 14, 123–133] and [Renyi, 1958, J. de Mathematique 37, 393–398] on general bounds using Borel functions of Bonferroni summations. This general method can be used to derive inequalities of specific forms. In particular, we use the proposed general method to re-derive the celebrated Dawson–Sankoff degree-two lower bound; develop a degree-three bound that is complementary in structure to the Sobel–Uppuluri–Galambos (degree-three) lower bound; and obtain a degree-three lower bound similar in structure to the Dawson–Sankoff bound. Numerical examples to illustrate are included.