I know it's true that if $(u_j)_{j\in\mathbb{N}}$ is a sequence of integrable functions in $\mathcal L^1$, $u_j\rightarrow u$ and $|u_j|\leq w, \forall j\in\mathbb N$ for some $w\in \mathcal L^1$ then this means that $$|u|=\lim_{j\rightarrow \infty}|u_j|\leq w$$ which implies that $u\in \mathcal L^1$.
Now lets move to the $\mathcal L^p$ space. Does this reasoning work as well?
Can I say that if $(u_j)_{j\in\mathbb{N}}$ is a sequence of integrable functions in $\mathcal L^p$, $u_j\rightarrow u$ and $|u_j|\leq w, \forall j\in\mathbb N$ for some $w\in \mathcal L^p$, then
$$(u_j^p)\in \mathcal L^1, \forall j\in\mathbb N$$ and then simply (given that $p\in[1,\infty)$) $$|u|^p=\lim_{j\rightarrow \infty}|u_j|^p\leq w^p$$ which implies that $u^p\in \mathcal L^1$ (which is the same as $u\in\mathcal L^p$) ?
Is it fair to say that any (convergent) sequence of functions in $\mathcal L^p$ that are bounded by a function $w\in\mathcal L^p$ converge in $\mathcal L^p$? I know that if the sequence is a Cauchy sequence then the latter is true because the space $\mathcal L^p$ is complete.
Thanks in advance.
$|u_j-u|^p2}\leq 2^{p} (|u_j|^{p}+|u|^{p}) \leq 2^{p} (|w|^{p}+|u|^{p})$. Noting that $\int |u|^{p} \leq \lim \inf \int |u_j|^{p}\leq \int |w|^{p}<\infty$ (where we have used Fatou's Lemma for the first inequality) it follows that $|u_j-u|^{p} \to 0$ so $u_j \to u$ in $L^{p}$.