Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / A / 2 / b / Solution

For events $E_n$, the Borel-Cantelli first lemma states that ${\sum }_n\mathbb{P}(E_n)<\infty$ implies $\mathbb{P}(E_n\text{ infinitely often})=0$. Indeed,These unions decrease to $\mathrm{limsup}{E}_n$, so continuity from above proves the assertion.

The Borel-Cantelli second lemma states that independent events with ${\sum }_n\mathbb{P}(E_n)=\infty$ satisfy $\mathbb{P}(E_n\text{ infinitely often})=1$. For every fixed $m$, independence and $1-x\le e^{-x}$ giveContinuity from above shows there is probability zero of no event after $m$. A countable union over $m$ gives probability zero of only finitely many occurrences. Independence is needed for this second conclusion, but not for the first.