Let $E \subset \mathbb R$ be Lebesgue measurable with $\mathcal L^1(E) < \infty$. Let $1 \le p < \infty$, and let $(f_n)_{n=1}^{\infty}$ be a sequence of Lebesgue-measurable maps $f_n: E \to \mathbb R$. Suppose there exists a constant $M \in [0,\infty)$ such that
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\begin{align*}
|f_n(x)| \le M
\end{align*}
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for every $n \in \mathbb N$ and every $x \in E$. Then $f_n \in L^p(E)$ for every $n \in \mathbb N$, and