Math 6040 • Probability

Answer the following two questions.
\begin{enumerate}

\item
Let \(\{X_{n}\}_{n \in \mathbb{N}}\) be independent, almost surely finite random variables. Show that
\[
\sup_{n} X_{n} < \infty \text{ a.s.}
\]
if, and only if, \(\exists c < \infty\) such that
\[
\sum_{n} P(X_{n} > c) < \infty.
\]



\item
Let \(\{A_{n}\}_{n \in \mathbb{N}}\) be a sequence of events with \(P(A_{n}) \ge \delta\) for each \(n\). Show that
\[
P(A_{n}\text{ happens infinitely often}) \ge \delta.
\]

\end{enumerate}