For (a), take
\(\{b_{k}(x)\}\subset \in C[a, b]\) such that
\(b_{k}(x)\to f(x)\) in
\(L^{p}[a, b]\) and
\(\{c_{k}(x)\} \subset \in C[a, b]\) such that
\(c_{k}(x)\to g(x)\) in
\(L^{p'}[a, b]\text{.}\) We show that
\(\{b_{k}(x)c_{k}(x)\}\) is Cauchy in
\(L[a, b]\) and that
\(f(x)g(x)\) is a realization of this sequence.
Note that there exists some \(M > 0\) such that for all \(k, l\)
\begin{equation*}
\Vert b_{l}(x) \Vert_{L^{p}[a,b]}, \Vert c_{k}(x)\Vert_{L^{p'}[a,b]}\le M.
\end{equation*}
Then
\begin{align*}
\amp \Vert b_{k}(x)c_{k}(x)-b_{l}(x)c_{l}(x)\Vert_{L[a,b]} \\
\le \amp \Vert \left(b_{k}(x)-b_{l}(x)\right)c_{k}(x)\Vert_{L[a,b]} +
\Vert b_{l}(x)\left( c_{k}(x)-c_{l}(x) \right)\Vert_{L[a,b]}\\
\le \amp \Vert \left(b_{k}(x)-b_{l}(x)\right)\Vert_{L^{p}[a,b]} \Vert c_{k}(x)\Vert_{L^{p'}[a,b]}
+ \Vert b_{l}(x) \Vert_{L^{p}[a,b]} \Vert \left( c_{k}(x)-c_{l}(x) \right)\Vert_{L^{p'}[a,b]}
\end{align*}
Since \(\Vert \left(b_{k}(x)-b_{l}(x)\right)\Vert_{L^{p}[a,b]} \to 0\) and \(\Vert \left( c_{k}(x)-c_{l}(x) \right)\Vert_{L^{p'}[a,b]}\to 0\) as \(k, l\to \infty\text{,}\) we can conclude that \(\Vert b_{k}(x)c_{k}(x)-b_{l}(x)c_{l}(x)\Vert_{L[a,b]} \to 0\) as \(k, l\to \infty\text{.}\) Since \(b_{k}(x)\to f(x)\) \(a.e.\) and \(c_{k}(x)\to g(x)\) \(a.e.\text{,}\) it follows that \(b_{k}(x)c_{k}(x)\to f(x)g(x)\) \(a.e.\text{,}\) making \(f(x)g(x)\) as a realization of the Cauchy sequence \(\{b_{k}(x)c_{k}(x)\}\text{.}\)
Finally, using
\begin{equation*}
\int_{a}^{b} |f(x)||g(x)|\, dx = \lim_{k\to \infty} \int_{a}^{b} |b_{k}(x)||c_{k}(x)|\, dx
\end{equation*}
and
\begin{equation*}
\int_{a}^{b} |f(x)|^{p}\, dx= \lim_{k\to \infty} \int_{a}^{b} |b_{k}(x)|^{p}\, dx
\end{equation*}
as well as
\begin{equation*}
\int_{a}^{b} |g(x)|^{p'}\, dx= \lim_{k\to \infty} \int_{a}^{b} |c_{k}(x)|^{p'}\, dx
\end{equation*}
it follows from the Hlder’s inequality
\begin{equation*}
\int_{a}^{b} |b_{k}(x)||c_{k}(x)|\, dx
\le \left(\int_{a}^{b} |b_{k}(x)|^{p}\, dx\right)^{1/p}
\left(\int_{a}^{b} |c_{k}(x)|^{p'}\, dx\right)^{1/p'}
\end{equation*}
that the same inequality holds for \(f\) and \(g\text{.}\)
For (b), the key is how the assumption \(|g(x)|\le M\) \(a.e.\) can be reflected in a Cauchy sequence from \(C[a, b]\) approximating \(g\) in \(L[a, b]\text{.}\) Let \(\{c_{k}(x)\}\subset C[a, b]\) be a Cauchy sequence for \(g\in L[a, b]\text{.}\) Define
\begin{equation*}
c^{M}_{k}(x)=\begin{cases} M \amp \text{ if } c_{k}(x) > M\\
c_{k}(x) \amp \text{ if } -M \le c_{k}(x)\le M\\
-M \amp \text{ if } c_{k}(x) < -M\\
\end{cases}
\end{equation*}
We claim that \(\{ c^{M}_{k}(x)\}\subset C[a, b]\) is a Cauchy sequence equivalent to \(\{c_{k}(x)\}\text{.}\)
First, observing that
\begin{equation*}
|c^{M}_{k}(x)-c^{M}_{l}(x)|\le |c_{k}(x)-c_{l}(x)|
\end{equation*}
holds point wise. It then follows that
\(\{ c^{M}_{k}\}\subset C[a, b]\) is a Cauchy sequence in
\(L[a, b]\text{.}\) Next, similar to the proof of
Proposition 1.2.16, we see that
\begin{equation*}
|c_{k}(x)-c^{M}_{k}(x)| =(c_{k}(x)-M)_{+}-(c_{k}(x)+M)_{-}
\end{equation*}
and that \((c_{k}(x)-M)_{+}=-(M-c_{k}(x))_{-}\to 0\) in \(L^{p}[a, b]\text{,}\) as \(M-g(x)\ge 0\) and \(M-c_{k}(x) \to M-g(x)\) in \(L^{p}[a, b]\text{;}\) similarly, \((c_{k}(x)+M)_{-} \to 0\) in \(L^{p}[a, b]\text{.}\) This shows that \(\{ c^{M}_{k}(x)\}\) is equivalent to \(\{c_{k}(x)\}\text{.}\)
Now if \(\{b_{k}(x)\}\subset C[a, b]\) is a Cauchy sequence for \(f\) in \(L[a, b]\text{,}\) then we claim that \(\{b_{k}(x) c_{k}^{M}(x)\} \) is a Cauchy sequence for \(f(x)g(x)\) in \(L[a, b]\text{.}\) This is because
\begin{align*}
\amp \int_{a}^{b}|b_{k}(x)c_{k}^{M}(x)-b_{l}(x)c_{l}^{M}(x)|\, dx \\
\le \amp \int_{a}^{b}|\left( b_{k}(x)- b_{l}(x)\right)c_{k}^{M}(x)|\, dx +
\int_{a}^{b}|b_{l}(x) \left(c_{k}^{M}(x)-c_{l}^{M}(x)\right)|\, dx\\
\le \amp M \int_{a}^{b}| b_{k}(x)- b_{l}(x)|\, dx +
\int_{a}^{b}|b_{l}(x) \left(c_{k}^{M}(x)-c_{l}^{M}(x)\right)|\, dx
\end{align*}
For any \(\epsilon > 0\text{,}\) first find \(N\) such that for all \(k, l\ge N\text{,}\) \(M \int_{a}^{b}| b_{k}(x)- b_{l}(x)|\, dx < \epsilon\text{.}\) Note that there exists some \(M' > 0\) such that \(\int_{a}^{b}|b_{l}(x)|\, dx \le M'\) for all \(l\text{,}\) and that there exists some \(\delta > 0\) such that for any open set \(G\) with \(|G| < \delta\text{,}\) we have \(2M \int_{G} |b_{l}(x)|\, dx < \epsilon/M'\) for all \(l\text{.}\) For such a \(\delta > 0\text{,}\) we can find some open set \(G\) with \(|G| < \delta\) such that a subsequence of \(\{c_{k}^{M}(x) \}\text{,}\) still denoted as \(\{c_{k}^{M}(x) \}\text{,}\) satisfies \(c_{k}^{M}(x) \to \phi(x)\) uniformly on \(G^{c}\text{.}\) Therefore, we can find some \(K\) such that for all \(k, l\ge K\text{,}\) we have \(| c_{k}^{M}(x)-c_{l}^{M}(x)| < \epsilon/M'\) for all \(x\in G^{c}\text{,}\) which leads to
\begin{equation*}
\int_{G^{c}} |b_{l}(x) \left(c_{k}^{M}(x)-c_{l}^{M}(x)\right)|\, dx\le \epsilon.
\end{equation*}
But
\begin{equation*}
\int_{G} |b_{l}(x) \left(c_{k}^{M}(x)-c_{l}^{M}(x)\right)|\, dx\le 2M \int_{G} |b_{l}(x) |\, dx < \epsilon.
\end{equation*}
This leads to
\begin{equation*}
\int_{a}^{b}|b_{k}(x)c_{k}^{M}(x)-b_{l}(x)c_{l}^{M}(x)|\, dx < 3\epsilon\text{.}
\end{equation*}
