Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use Bochner integral. I'm trying to prove below result, i.e.,
Theorem: Let $p, q \in [1, \infty)$. Let $f,f_n \in \mathcal L_p (X, \mu, E)$ and $g,g_n \in \mathcal L_q (X, \mu, E)$ such that $f_n \to f$ a.e. and $g_n \to g$ a.e. Assume $|f_n|^p \le |g_n|^q$ a.e. for all $n$ and $\|g_n\|_q \xrightarrow{n \to \infty} \|g\|_q$. Then $\|f_n -f\|_p \xrightarrow{n \to \infty} 0$.
Could you have a check on my attempt?
Proof It follows from $|f_n|^p \le |g_n|^q$ a.e. for all $n$ that $|g|^q - |f|^p\ge 0$ a.e. So $\int (|g|^q - |f|^p)$ is well-defined in the extended non-negative real numbers. We have $$ \begin{align*} \int (|g|^q - |f|^p) &= \int \liminf_n (|g_n|^q - |f_n|^p) \\ &\le \liminf_n \int (|g_n|^q - |f_n|^p) \quad \text{by Fatou's lemma}\\ &\overset{(\star)}{=} \int|g|^q - \limsup_n \int |f_n|^p, \end{align*} $$ and $$ \begin{align} \int (|g|^q+|f|^p) &= \int \liminf_n (|g_n|^q+|f_n|^p) \\ &\le \liminf_n \int (|g_n|^q+|f_n|^p) \quad \text{by Fatou's lemma} \\ &\overset{(\star)}{=} \int |g|^q + \liminf_n \int |f_n|^p, \end{align} $$ where $(\star)$ follows from below lemma, i.e.,
Lemma Let $a_n, b_n \in \mathbb R$ for all $n$. If $(a_n)$ is convergent, then $\liminf (a_n+b_n) = \lim a_n + \liminf b_n$.
It follows from $\int|g|^q$ is finite that $$ \limsup_n \int |f_n|^p \le \int |f|^p \le \liminf_n \int |f_n|^p. $$
As such, $\lim_n \|f_n\|_p = \|f\|_p$. The claim then follows from below Lemma, i.e.,
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 $\|f_n - f\|_p \to 0$.
