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\vspace*{.3in}
\noindent{\bf Beals, page 25 \#5:}
Set $g_n(x) = \inf_{j\geq n} f_j(x)$.  Then for all $n\in\N$ and $x\in\R$,
\begin{itemize}
\item $\liminf f_n(x) = \lim g_n(x)$,
\item $0 \leq g_n(x) \leq f_n(x)$,
\item $g_{n+1}(x) \geq g_n(x)$
\end{itemize}
The second and third items allow us to apply the monotone convergence
theorem to $g_n$ and obtain
$$
\int \liminf f_n = \int \lim g_n = \lim \int g_n = \liminf \int g_n
\leq \liminf \int f_n.
$$
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