Let given that {A_i}_i elementof I is a countable collection of events Now consider P (bigcap_i elementof I A_i) = 1 - P ((bigcap_i elementof I A_i)^c) - 1 Now consider P((bigcap_i elementof I A_i)^c) = P (bigcup_i elementof I A^c_i) lessthanorequalto Sigma_i elementof I P(A^c_i) Rightarrow P((bigcap_i elementof I A_i)^c) lessthanorequalto Sigma_i elementof I P(A^c_i) [sub - activity] so - P ((bigcap_i elementof I A_i)^c) greaterthanorequalto - Sigma_i elementof I P (A^c_i) - 2 From 1 and 2 we have P(bigcap_i elementof I A_i) greaterthanorequalto 1 - Sigma_i elementof I P(A^c_i)