I've been given this problem statement:
Let $f_n$, $\phi \in L^p(\Omega)$, $p\in[1,\infty)$ s.t.:
$|f_n(x)|\leq\phi(x) $ a.e. in $\Omega$
$f_n(x) \rightarrow f(x)$ a.e. in $\Omega$
Prove that $f\in L^p$
By the Lebesgue dominated convergence theorem, it can be seen that these conditions imply $\int_{\Omega} f_i \rightarrow \int_{\Omega} f \leq \int_{\Omega} \phi$. However, we need $||f||_p<+\infty $. Is this true because $|f|\leq\phi \implies |f|^p\leq|\phi|^p\implies \int_{\Omega}|f|^p\leq\int_{\Omega}|\phi|^p <+\infty$? Or do I need to show this last line somehow?
Thank you for any feedback.