Solution
= Solution
The <Cauchy-Schwarz inequality> gives
$$
\mathbb E|ZX_n|
\leq(\mathbb E|Z|^2)^{1/2}
(\mathbb E|X_n|^2)^{1/2}
\leq\lVert Z\rVert_2<\infty.
$$
Thus \b[$ZX_n$ is integrable] for every $n$.
= Solution
The <Cauchy-Schwarz inequality> gives
$$
\mathbb E|ZX_n|
\leq(\mathbb E|Z|^2)^{1/2}
(\mathbb E|X_n|^2)^{1/2}
\leq\lVert Z\rVert_2<\infty.
$$
Thus \b[$ZX_n$ is integrable] for every $n$.