It would be great if someone could name me a reference where I can find a proof for the following statement, thank you! :)
Given is a bounded sequence $(f_{n})_{n\in\mathbb{N}}\in L^{\infty}$. Then there exists a $f\in L^{\infty}$ and a subsequence $f_{n_{k}}$ of $f_{n}$ that convergeces weakly* to $f$ in $L^{\infty}$.
You did not specify the basic measure space but I will assume that $L^{1}$ is separable and you are looking at $L^{\infty}$ as the dual of $L^{1}$. In this case Banach Alaoglu Theorem tells you that any closed ball in $L^{\infty}$ is compact in weak* topology. Further separability if $L^{1}$ makes this ball metrizable. Hence every sequence in it has a weak* convergent subsequence.