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Theorem brings the classical Borel-Cantelli lemma much closer to the central limit theorem and law of the iterated logarithm, without any additional assumptions concerning the divergence of the sums of the variances of the random variables in question, assumptions quite essential in both latter results. It sharpens Levy’s conditional form of the Borel-Cantelli lemma [5, Corollary 68, p. 249], and an improved version due to Dubins and Freedman ([2, Theorem 1] or [6, Corollary VII-2-6, p. 152]) which is stated

A stronger form of the Borel-Cantelli Lemma