Show $\|f\|_p\leq \lim\inf\|f_n\|$

Question

$\Omega$ is a bounded domain of $\mathbb R^n$. If $\{f_n\}\subset L^p(\Omega)$ and $f_n\rightarrow f\in L^p(\Omega)$ weakly, then
$$\|f\|_p\leq \lim_{n\rightarrow\infty}\inf\|f_n\|$$

Answer 1

You are asking to prove why $$\left(\int |f|^p\right)^{1/p} \leq \lim_{n\rightarrow \infty} \inf \left(\int {|f_n|^p}\right)^{1/p},$$

so it is enough to prove that $\left(\int |f|^p\right)\leq \lim_{n\rightarrow \infty} \inf \left(\int {|f_n|^p}\right)$. Now if $f_n \to f$ pointwise a.e. then $|f_n|^p \to |f|^p$ pointwise a.e. Thus $$\begin{eqnarray*} \int |f|^p &=& \int \lim_{n \rightarrow \infty} |f_n|^p \\ &=&\int \lim_{n \rightarrow \infty} \inf |f_n|^p \\ &\leq& \lim_{n \rightarrow \infty} \inf \int |f_n|^p \end{eqnarray*}$$

where the last step was made using Fatou's lemma. Q.E.D.