$||f||_p$ bounded by $\lim\inf ||f_n||_p$

Question

(True/False)
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
If $f_n\in L^p(\Omega)$ for $1<p<\infty$ and converges weakly to $f\in L^p$, then $||f||_p\leq \lim\inf_{n\to\infty}||f_n||_p$

This is a practice qual problem.

I'm a bit stuck on how to use the condition that $f_n\to f$ weakly.
$\forall g\in L^q$, $\int_\Omega (f_n - f)g\to 0$
If we let $g = \chi_\Omega$, then $\int_\Omega (f_n -f) \to 0$, but this doesn't imply convergence in $L^1$

If we can show $||f||^p_p\leq \int_\Omega \lim\inf |f_n|^p$, then we would be done by fatou's lemma, but I don't see how weak convergence implies this condition.

Answer 1

Hint: start with Hölder $$ \left|\int_{\Omega}f_n g\right|\le\|f_n\|_p\|g\|_q,\quad \forall g\in L^q. $$ Now take $\liminf$ of both sides and note that $\liminf=\lim$ if the latter exists. Use the weak convergence and the norm definition.