Convergence in an $L^p$ space
Solution 1:
This question has been answered many many times for $p \geq 1$. For $0<p<1$ this follows from Fatou's Lemma: $|f-f_n|^{p} \leq |f|^{p}+|f_n|^{p}$ in this case so $|f|^{p}+|f_n|^{p} -|f-f_n|^{p} $ is non-negative Fatou's Lemma applied to this sequence gives the result immediately.