The Borel-Cantelli Lemma is a key result showing that random variables convergence almost surely. Intuitively, the result is linked to the concept of events occurring infinitely often, i.e. for some event \(A_n\) in \((\Omega, \mathcal{F}, \mathbb{P})\) we have \[ \forall k>0,\exists n>k~s.t.~\omega \in A_{n}. \] Recall from the definition of Limit Supremum & Limit Infimum that for an infinite sequence of events \(A_{1}, A_{2}, \dots\) on \((\Omega,\mathcal{F}, \mathbb{P})\), \[ \omega \in A^\infty\equiv\limsup_{n \to \infty }A_{n}=\bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_{m}= \omega \in A_{n}\text{ for infinitely many values }n=A_{n}\text{ occurs i.o.} \] The idea of limsup is that we are considering a smaller and smaller portion of the tail of the sequence.
Let \(\{A_{n}, n \geq 1\}\) be an infinite sequence of events from some probability space \((\Omega, \mathcal{F}, \mathbb{P})\). 1. If \(\sum_{n=1}^\infty \mathbb{P}(A_{n})<\infty\). then \(\mathbb{P}(A_{n}~i.o.)=0\). 2. If the events \(A_{n}\) are independent and \(\sum_{n=1}^\infty \mathbb{P}(A_{n})=\infty\) then \(\mathbb{P}(A_{n}~i.o.)=1\).
Proof: For (1), we have that \(\{B_{n}=\cup_{k=n}^\infty A_{k}\}\searrow A^\infty\) as \(n \to \infty\), thus \[ 0\leq \mathbb{P}(A^\infty)\leq \mathbb{P}(\cup_{k=n}^\infty A_{k})\leq \sum_{k=n}^\infty \mathbb{P}(A_{k})\to 0\quad (n\to \infty). \] For (2) we recall that \(\{ A_{n}~i.o. \}=\cap_{n=1}\cup_{k=n}^\infty A_{k}=\cap_{n=1}B_{n}\). To show that \(\mathbb{P}(\{ A_{n}~i.o. \})=\mathbb{P}(\cap_{n=1}B_{n})=1\) we equivalently show the converse, i.e. \(\mathbb{P}(\cup_{n=1}B_{n}^c)=1-\mathbb{P}(\cap_{n=1}B_{n})=0\), but we note that \(0\leq \mathbb{P}(\cup_{n=1}B_{n}^c)\leq \sum_{n=1} ^\infty \mathbb{P}(B_{n}^c)\). To complete the proof, we take \(N>n\) (to have a finite intersection) and consider \[ \mathbb{P}(\cap_{k=n}^NA_{k}^c)=\prod_{k=n}^N\mathbb{P}(A_{k}^c)=\prod_{k=n}^N(1-\mathbb{P}(A_{k})). \] Using the inequality \(1-x\leq e^{-x}\) and the identity \(e^ae^b=e^{a+b}\) we write \[ \mathbb{P}(\cap_{k=n}^NA_{k}^c)\leq \prod_{k=n}^Ne^{-\mathbb{P}(A_{k})}=e^{-\sum_{k=n}^N\mathbb{P}(A_{k})}\stackrel{N \to \infty}{\to}e^{-\sum_{k=n}^\infty \mathbb{P}(A_{k})}=e^{-\infty}=0, \] by condition of the lemma. \(\square\)