Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 34 / 2 / a / Solution

For events $A_n$, define their limit superior byIt is the event that infinitely many $A_n$ occur. The two Borel-Cantelli lemmas are as follows: if ${\sum }_n\mathbb{P}(A_n)<\infty$, then $\mathbb{P}(A_{\infty })=0$, without any independence assumption; if the $A_n$ are independent and ${\sum }_n\mathbb{P}(A_n)=\infty$, then $\mathbb{P}(A_{\infty })=1$.

For the first assertion, the union bound gives, for every $N$,The tail of a convergent nonnegative series tends to zero, proving the first Borel-Cantelli lemma.

For the second assertion, independence and $1-u\le e^{-u}$ for $0\le u\le 1$ giveBy continuity of probability, $\mathbb{P}({\bigcap }_{n\ge N}{A}_n^c)=0$. The event that only finitely many $A_n$ occur is the countable union of these zero-probability events over $N$. Its probability is zero. This proves the second Borel-Cantelli lemma, and hence both requested assertions: