Let $({f_n})_{n\geq 1}$ be a sequence of functions on $[0,1]$ such that $\lim _{n\rightarrow \infty}f_{n}(x)=f(x)$ almost everywhere with respect to Lebesgue measure and $$\sup_{n\geq 1}||f_n||_{L^4}<\infty.$$
Prove that $$\lim_{n\rightarrow \infty}||f_n - f||_{L^3} = 0.$$
Since we're on a finite measure space I see how we get that $f_n \in L^3$. But I don't see how I can use a convergence theorem on $L^p$ larger than one.