Math 6010 • Linear Models

Suppose \(\{E_1,E_2,\ldots\}\) is a sequence of measurable (otherwise arbitrary) events. Suppose \(\{F_1,F_2,\ldots\}\) is another sequence of measurable events also totally independent of all of the \(E_i\)'s. Assume the following:
\begin{parts}
\item \(P(E_k\text{, infinitely often})=1\);
\item there exists some \(p>0\), such that for all \(k\geq 1\), \(P(F_k)\geq p\).
\end{parts}
Prove:
\[
P(E_k\cap F_k\text{, infinitely often})\geq p>0.
\]
(10 points)