Let $(X,M,\mu)$ be a finite measure space and $f_n\in L^2(\mu)$ s.t $\|f_n\|_2\leq M$ and $f_n\to f$ a.e. Show that $f\in L^2(\mu)$ and $f_n\to f$ in $L^1$.
So I was able to prove that $f\in L^2(\mu)$. Moreover, I used Holder's inequality to show that $f_n,f\in L^1(\mu)$ and also $\|f_n\|_1,\|f\|_1\leq M\sqrt{\mu(X)}$. Now I want to show that $\int_X|f_n-f|d\mu\to 0$. Let $\varepsilon>0$.
Now I used Egoroff's theorem to show that there exists $E\subset X$ s.t $\mu(E)<\varepsilon$ and $f_n\to f$ uniformly on $E^c$.
So I tried bounding the integral $\int_X|f_n-f|d\mu=\int_E|f_n-f|d\mu+\int_{E^c}|f_n-f|d\mu$. Bounding the $\int_{E^c}|f_n-f|d\mu$ is easy because of uniform convergence, but I wasn't able to say anything smart about $\int_E|f_n-f|d\mu$ (I know it is bounded by $2M\sqrt{\mu(X)}$ but this doesn't help me).
Any hint would be appreciated.
Notice that $\Vert f_n-f\Vert_2$ is bounded by $M'>0$. Then
$$ \int_E \vert f_n-f\vert d\mu\leq \Vert f_n-f\Vert_2\cdot \Big( \int_X \mathbf{1}_E^2d\mu \Big)^{\frac{1}{2}} =\Vert f_n-f\Vert_2\cdot \sqrt{\mu(E)}.$$