Let $1< p < \infty$, $\{f_n\}_n \subset L^p[0,1]$ s.t. $f_n:[0,1] \to \mathbb{R}$, $f_n \to f$ a.e., and $||f_n||_{L^p} \leq M < \infty$ for all $n$. Then, given $1 \leq q < p$, we want to show that $f_n \to f$ in $L^q$.
I think that you can show this result by using the Rellich-Kondrachov theorem (i.e. $L^p \hookrightarrow L^q$ compactly for $q < p$) to extract a convergent subsequence in $L^q$. Then you can use the fact that $f_n \to f$ a.e. to show that the whole sequence must converge to $f$ in $L^q$.
However, I was wondering if this approach might be a little overkill, and if I should be using some less overpowered tools to prove the statement (assuming the argument above is valid).