If $2^p(\vert f_n \vert^p+\vert f \vert^p)-\vert f_n-f \vert^p \geq 0$ then how would I show $$2^{p+1} \int \vert f \vert^p=\int \lim_{n\to \infty}[2^p(\vert f_n \vert^p+\vert f \vert^p)-\vert f_n \vert^p].$$
This I found while proving:
Let {$f_n$ } be a sequence in $L^p,1\leq p <\infty,$ such that $f_n \rightarrow f $ a.e and $f \in L^p$.If $\lim_{n\to \infty}||f_n||_p = ||f||_{p}$,then $\lim_{n\to \infty}||f_n-f||_{p}=0$
I wanted to know how it happened? Need help!!!