# Confirmation of convergence in subdomain

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 $$A \subset \Omega$$, $$\int_\Omega \lvert f \rvert^p = \int_A \lvert f \rvert^p + \int_{\Omega\setminus A} \lvert f \rvert^p \geq \int_A \lvert f \rvert^p.$$