# P[limsup] of independent events

1. Aug 14, 2009

### student12s

1. The problem statement, all variables and given/known data

Let (A_n) be a sequence of independent events such that Pr(A_n)<1 for all n. Show that P[limsup A_n]=1 if and only if P(\cup_{n=1}^{+\infty} A_n)=1

2. Relevant equations

3. The attempt at a solution
Suppose P[\limsup A_n]=1. Define $B_k=\cup_{i=k}^{+\infty} A_i so that B_n\downarrow \limsup_{n\to+\infty} A_n. Thus \lim_{n\to+\infty} P(B_n)=P(\limsup_{n\to+\infty} A_n)=1. So, for all \epsilon>0, there is M so that 1\geq P(\cup_{i=1}^{+\infty} A_i)\geq P(\cup_{i=M}^{+\infty}A_i)\geq 1-\epsilon$, so P(\cup_{i=1}^{+\infty} A_i)=1.
I am having trouble with the other direction. Any hints?