If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$

You are on the right track. By Egoroff's, it is enough to show that for an arbitrary small set $A$ we have $$\int_A |f_n|^p dm \leq \epsilon.$$

By Holders, for $1/q+1/q'=1$, we get $$ \int_A |f_n|^p \cdot 1 \leq \left( \int_A |f_n|^{pq} dm \right)^{1/q} \left( \int_A 1^{q'} dm \right)^{1/q'}.$$

Choose $pq=2$ so that $1/q'=1-1/q=1-p/2=(2-p)/2$ or $q'=2/(2-p)$. Then $$\int_A |f_n|^p dm \leq M^{1/q} m(A)^{1/q'}.$$

The rest follows easily.

NOTES: The idea is to remember you can use Holder with the function $g=1$ and that somehow you want to convert the power of $p$ to a power of $2$. Holder requires the exponents to be slightly greater than 1, so $p<2$ makes sense. In addition, note that $q' \leq 0$ for $p \geq 2$, which does not allow for the same bound.