Strong convergence means $\|f_j-f\|_p\to0$; weak convergence means $\int f_jh\to\int fh$ for every $h\in L^q$. By duality choose $h$ of norm one with $\int fh$ arbitrarily close to $\|f\|_p$. <Holder inequality> and passage to the limit give $\|f\|_p\leq\liminf\|f_j\|_p$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 23H / a / Solution

 Solution (source code)