I'm trying to prove this generalized version of dominated convergence theorem for Banach space. Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space, $(E, |\cdot|)$ a Banach space, and $p \in [1, \infty)$.
Theorem: Let $f,g,f_n, g_n \in \mathcal L_p (X, \mu, E)$ such that $f_n \to f$ $\mu$-a.e., $g_n \to g$ $\mu$-a.e., $|f_n| \le |g_n|$ $\mu$-a.e., and $\|g_n\|_p \to \|g\|_p$. Then $\|f_n -f\|_p \to 0$.
Could you have a check on my attempt?
My proof: Because $p \ge 1$ and $|g_n| \ge |f_n|$ $\mu$-a.e., we have $|g_n|^p - |f_n|^p \ge 0$ $\mu$-a.e. By Fatou's lemma, $$ \begin{align} \int (|g|^p - |f|^p) &= \int \liminf_n (|g_n|^p - |f_n|^p) \\ &\le \liminf_n \int (|g_n|^p - |f_n|^p) \\ &= \lim_n \int |g_n|^p - \limsup_n \int |f_n|^p. \end{align} $$
Hence $$ \limsup_n \int |f_n|^p \le \int |f|^p. $$
We have $|g_n|^p+|f_n|^p \ge 0$ everywhere. By Fatou's lemma, $$ \begin{align} \int (|g|^p+|f|^p) &= \int \liminf_n (|g_n|^p+|f_n|^p) \\ &\le \liminf_n \int (|g_n|^p+|f_n|^p) \\ &= \lim_n \int |g_n|^p + \liminf_n \int |f_n|^p. \end{align} $$
Hence $$ \liminf_n \int |f_n|^p \ge \int |f|^p. $$
It follows that $$ \liminf_n \int |f_n|^p \ge \int |f|^p \ge \limsup_n \int |f_n|^p. $$
As such, $$ \lim_n \|f_n\|_p = \|f\|_p. $$
Lemma: Let $f_n, f \in \mathcal L_p (X, \mu, E)$ such that $f_n \to f$ $\mu$-a.e. and $\|f_n\|_p \to \|f\|_p$ as $n \to \infty$. Then $\|f_n - f\|_p \to 0$.
The claim then follows from above Lemma.
