A Beautiful Probability Theorem

We all know that, given two events A and B, the probability of the union A U B is given by the formula P(A U B) P(A) P(B) - P( AB) where AB represents the intersection of A and B. Most of us even know that It generalizes to n independent events, and this formula is known as the inclusion-exclusion principle. Let us consider n events A(1), A(2), ..., A(n) where A(k) is for a positive integer number, the property to be divisible by the square of the k-th prime number. We assume here that the first prime number is 2. These events are independent because we are dealing with prime numbers. As n tends to infinity, 1 - P( A(1) U A(2) U ... U A(n)) tends to the probability, for a positive integer number, to be square-free.