related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states: If the events En are independent and the sum of the probabilities of the En diverges to infinity, then the probability that infinitely many of them occur is 1. That is: The assumption of independence can be weakened to pairwise independence, but in that case the proof is more difficult. Example The infinite monkey theorem, that endless typing at random will, with probability 1, eventually produce every finite text (such as the works of Shakespeare), amounts to the statement that a (not necessarily fair) coin tossed infinitely often will eventually come up Heads. This is a special case of the second Lemma. The lemma can be applied to give a covering theorem in Rn. Specifically , if Ej is a collection of Lebesgue measurable subsets of a compact set in Rn such that then there is a sequence Fj of translates such that apart from a set of measure zero. Proof Suppose that and the events are independent. It is sufficient to show the event that the En's did not occur for infinitely many values of n has probability 0. This is just to say that it is sufficient to show that Noting that: , it is enough to show: . Since the are independent: The convergence test for infinite products guarantees that the product above is 0, if diverges. This completes the proof. Counterpart Another related result is the so-called counterpart of the Borel–Cantelli lemma. It is a counterpart of the Lemma in the sense that it gives a necessary and sufficient condition for the limsup to be 1 by replacing the independence assumption by the completely different assumption that is monotone increasing for sufficiently large indices. This Lemma says: Let be such that , and let denote the complement of . Then the probability of infinitely many occur (that is, at least one occurs) is one if and only if there exists a strictly increasing sequence of positive integers such that This simple result can be useful in problems such as for instance those involving hitting probabilities for stochastic process with the choice of the sequence usually being the essence. Kochen–Stone Let be a sequence of events with and then there is a positive probability that occur infinitely often. See also Lévy's zero–one law Kuratowski convergence Infinite monkey