I need to show that if we have a sequence $\{ f_n\}$ uniformly bounded in $L^{\infty}(\Omega)$, and $f_n \to f$ in $L^p_{\operatorname{ loc} }(\Omega)$ for some $p \ge 1$ and $\Omega$ be undounded subset of $\mathbb{R}^m$, then $f_n \to f$ in $L^q_{\operatorname{ loc}}(\mathbb{R}^m)$ for all $q \in [1, \infty)$
I could see that this is true for all $q\le p$ using Holder's inequality. How can I prove for $p \le q$.
Assume that $p\leqslant q$. According to this post, it suffices to prove that if $C\subset\mathbb R$ is compact, then $\int_C\left\lvert f_n(x)-f(x)\right\rvert^q\mathrm dx \to 0$. To this aim, observe that $$\left\lvert f_n(x)-f(x)\right\rvert^q=\left\lvert f_n(x)-f(x)\right\rvert^p \left\lvert f_n(x)-f(x)\right\rvert^{q-p} \\ \leqslant \left\lvert f_n(x)-f(x)\right\rvert^p2^{q-p}\left( \left\lvert f_n(x)\right\rvert^{q-p} +\left\lvert f(x)\right\rvert ^{q-p}\right) \leqslant \left\lvert f_n(x)-f(x)\right\rvert^p 2^{q-p} \sup_{t\in\mathbb R}\left\lvert f_n(t)\right\rvert^q +\left\lvert f(t)\right\rvert^q .$$ It is actually the case $p\gt q$ which follows from Hölder's inequality.