Convergence in $L^p(\Omega)$ implies convergence of integral

Question

The Hölder inequality for Lebesgue spaces is given as: \begin{align*} || f g ||_{L_1(\Omega)} \leq ||f||_{L_p(\Omega)} ||g||_{L_{p'}(\Omega)} \end{align*} where $p, p'$ are the dual exponents. Suppose $\Omega$ is bounded and that $f_j \to f$ in $L_p(Ω)$. Using Hölder’s inequality prove that \begin{align*} \int_{\Omega} f_j(x) \mathrm{d}x \rightarrow \int_{\Omega} f(x) \mathrm{d}x \ \ \text{as} \ \ j \rightarrow \infty \end{align*}

Answer 1

Use the inequality $\|FG\|_{L^1(\Omega)} \le \|F\|_{L^p(\Omega)}\|G\|_{L^{p'}(\Omega)}$ with $F = f_j - f$ and $G = 1$ to get $$\|f_j - f\|_{L^1(\Omega)} \le |\Omega|^{1/{p'}}\|f_j - f\|_{L^p(\Omega)}.$$

Then use the assumption $f_j \to f$ in $L^p(\Omega)$ to conclude.