Let $E\subset \mathbb{R}^n$ be measurable with $\mu(E)<\infty$. Given $p\in[1,+\infty[$ consider $\{f_n\}_{n\in\mathbb{N}}$ a bounded succession in $L^p(E)$ such that $f_n(x)\rightarrow f(x)$ for a.e. $x\in E$.
1) Prove that $f\in L^p(E)$.
2) If $1<p<\infty$, prove that $f_n\rightarrow f$ in $L^q(E)$ for all $q\in[1,p[$.
I think I have solved the first point but I can't figure out the second one for which I thought to write $E_n=\{x:\;|f_n(x)-f(x)|\geq \epsilon\}$ so that \begin{equation} \int_E|f_n-f|^q=\int_{E\backslash E_n} |f_n-f|^q+\int_{E_n}|f_n-f|^q\leq \mu(E\backslash E_n)\epsilon^q+\int_{ E_n} |f_n-f|^q \end{equation} but I can't prove that the second term tends to zero.
Thanks to the suggestion provided by Thomas, I have finally figured it out.
For $q\in[1,p)$,
\begin{equation} \int_{E_n} 1\cdot|f_n-f|^q\leq\mu(E_n)^{\frac{p-q}{p}}\cdot \left(\int_{E_n}|f_n-f|^p\right)^{\frac{q}{p}}\leq\mu(E_n)^{\frac{p-q}{p}}\cdot ||f_n-f||^q_{L^p(E)}\rightarrow 0. \end{equation}