Assume we have a sequence of functions $(f_n)_{n \ge 1} \in L^p$, where $L^p$ is the space of all measurable functions with a finite $p$-th moment, which almost surely converge to $f \in L^p$. Further we have that \begin{align*} \lim\limits_{n \rightarrow \infty} \| f_n\|_p =\|f\|_p \end{align*} This is sufficient to claim that $\|f_n - f \|_p \rightarrow 0$.
I have searched for a while on german and english sites for the proof of this statement, but wasn't succesful. It would be great, if someone could link me a source of the proof or write down his own.
