# 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 is
strictly larger than $j(1)$. Thus we can produce an infinite^{}
sequence 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 the
hypothesis^{} that ${\sum}_{i=1}^{\mathrm{\infty}}P({A}_{i})$ is finite. Therefore, $P(E)=0$.

Title | proof of Borel-Cantelli 1 |
---|---|

Canonical name | ProofOfBorelCantelli1 |

Date of creation | 2013-03-22 14:28:16 |

Last modified on | 2013-03-22 14:28:16 |

Owner | kshum (5987) |

Last modified by | kshum (5987) |

Numerical id | 6 |

Author | kshum (5987) |

Entry type | Proof |

Classification | msc 60A99 |