Suppose $\text{sup}\|f_n\|_p <+ \infty$ and $f_n \to f$ almost everywhere on $E$. I want to prove weak convergence $f_n \rightharpoonup f.$
I already have a proof that there exists a sub-sequence $\{f_{n_i}\}$ such that $f_{n_i}\rightharpoonup f$ weakly in $L^p.$
So in can reformulate my question as:
How to show from the weak convergence of such sub-sequence the convergence of the whole sequence?
You have to show that $\int f_ng \to \int fg$ for all $g \in L^{q}$ where $\frac 1 p+\frac 1 q=1$. For this it is enough to show that every subsequence of $(\int f_ng )$ has a subsequence converging to $\int fg$.
A sequence of real numbers $(a_n)$ converges to a real number $a$ iff every subsequence of $(a_n)$ has a further subsequence converging to $a$. [You can give a proof by contradiction].