# proof of Borel-Cantelli 1

Let $B_{k}$ be the event $\cup_{i=k}^{\infty}A_{i}$ for $k=1,2,\ldots,$. If $x$ is in the event $A_{i}$’s i.o., then $x\in B_{k}$ for all $k$. So $x\in\cap_{k=1}^{\infty}B_{k}$.

Conversely, if $x\in B_{k}$ for all $k$, then we can show that $x$ is in $A_{i}$’s i.o. Indeed, $x\in B_{1}=\cup_{i=1}^{\infty}A_{i}$ means that $x\in A_{j(1)}$ for some $j(1)$. However $x\in B_{j(1)+1}$ implies that $x\in A_{j(2)}$ for some $j(2)$ that is strictly larger than $j(1)$. Thus we can produce an infinite sequence of integer $j(1) such that $x\in A_{j(i)}$ for all $i$.

Let $E$ be the event $\{x:\,x\in A_{i}\mbox{ i.o.}\}$. We have

 $E=\bigcap_{k=1}^{\infty}\bigcup_{i=k}^{\infty}A_{i}.$

From $E\subseteq B_{k}$ for all $k$, it follows that $P(E)\leq P(B_{k})$ for all $k$. By union bound, we know that $P(B_{k})\leq\sum_{i=k}^{\infty}P(A_{i})$. So $P(B_{k})\rightarrow 0$, by the hypothesis that $\sum_{i=1}^{\infty}P(A_{i})$ is finite. Therefore, $P(E)=0$.

Title proof of Borel-Cantelli 1 ProofOfBorelCantelli1 2013-03-22 14:28:16 2013-03-22 14:28:16 kshum (5987) kshum (5987) 6 kshum (5987) Proof msc 60A99