Let $\{u_{n}\}_{n\in\mathbb{N}} \subset L^{p}(\Omega)$ be a sequence of function such that $u_{n} \to u$ in $L^{p}(\Omega)$ for $\Omega\subset \mathbb{R}^{N}$ bounded domain. I want to show that for any set $A \subset \Omega$, then, $u_{n}\to u$ in $L^{p}(A)$.

This is my attempt.
$||u_{n}-u||_{L^{p}(A)}\leq ||u_{n}-u||_{L^{p}(\Omega)}\to 0$ as $n\to\infty$.

My question : is it that simple? I am afraid I am doing something wrong in a very subtle way.

Any help or feedback is much appreciated!


I believe it is that simple as, for any measurable $A \subset \Omega$, $$\int_\Omega \lvert f \rvert^p = \int_A \lvert f \rvert^p + \int_{\Omega\setminus A} \lvert f \rvert^p \geq \int_A \lvert f \rvert^p.$$


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