TITLE: Can someone explain the Borel-Cantelli Lemma?
QUESTION [25 upvotes]: I’m looking for an informal and intuitive explanation of the Borel-Cantelli Lemma. The symbolic version can be found here.
What is confusing me is what ‘probability of the limit superior equals $ 0 $’ means.
Thanks!

REPLY [13 votes]: Here is a proof I wrote up for sci.math, but never posted:

Borel-Cantelli: Suppose that $\sum_{i=1}^\infty P(A_i)$ is finite, then the probability that infinitely many of the $A_i$ occur is $0$.
Proof:
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 $j(1)< j(2)< j(3)<\ldots$ 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
$$
E = \bigcap_{k=1}^\infty \bigcup_{i=k}^\infty A_i\tag{1}
$$
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, $P(E)=0$.

For events, $\bigcap\approx\inf$ and $\bigcup\approx\sup$. Since
$$
\limsup_{n\to\infty}a_n=\operatorname*{\inf\vphantom{p}}_{n\ge1}\sup_{k\ge n}a_k\tag{2}
$$
it makes sense to call
$$
\limsup_{n\to\infty}A_n=\bigcap_{n\ge1}\bigcup_{k\ge n} A_k\tag{3}
$$