proof of Borel-Cantelli 1

Let $B_k$ be the event ${\cup }_{i=k}^{\mathrm{\infty }}{A}_i$ for$k=1,2,\mathrm{\dots },$. If $x$ is in the event $A_i$’s i.o., then $x\in B_k$ for all $k$. So $x\in {\cap }_{k=1}^{\mathrm{\infty }}{B}_k$.

Conversely, if $x\in B_k$ for all $k$, then we can show that $x$is in $A_i$’s i.o. Indeed, $x\in B_1={\cup }_{i=1}^{\mathrm{\infty }}{A}_i$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 isstrictly larger than $j(1)$. Thus we can produce an infinitesequence of integer such that $x\in A_{j(i)}$ for all $i$.

Let $E$ be the event $\{x:x\in A_i\text{ i.o.}\}$. We have

$$E=\bigcap _{k=1}^{\mathrm{\infty }}\bigcup _{i=k}^{\mathrm{\infty }}{A}_i.$$

From $E\subseteq B_k$ for all $k$, it follows that $P(E)\le P(B_k)$ for all $k$. By union bound, we know that $P(B_k)\le {\sum }_{i=k}^{\mathrm{\infty }}P(A_i)$. So $P(B_k)\to 0$, by thehypothesis that ${\sum }_{i=1}^{\mathrm{\infty }}P(A_i)$ is finite. Therefore, $P(E)=0$.