I've been struggling with the following, and can't seem to determine a good direction to follow:
Let $f_n,f\in L^p$ where $f_n\rightarrow f$ almost everywhere and $\infty>p\geq1$. Then $\underset{n\rightarrow \infty}{\lim}\Vert f_n-f\Vert_p=0$ if and only if $\underset{n\rightarrow \infty}{\lim}\Vert f_n\Vert_p=\Vert f\Vert_p$.
Assuming that there is $L^p$ convergence it is easy to show that $\Vert f_n\Vert\rightarrow \Vert f\Vert$. However I've been stuck on the other direction, attempting to construct a dominating function (but failing). It seems to me that there should be a simple answer I'm overlooking.
In order to obtain an upper bound for $|f_n-f|^p$, by convexity of the function $t\mapsto |t|^p$, $$|f_n-f|^p\leq 2^{p-1}(|f_n|^p+|f|^p) $$ Thus $g_n:=2^{p-1}(|f_n|^p+|f|^p)-|f_n-f|^p$ is non-negative. Now you can apply Fatou's lemma on $\left\{g_n\right\}$.