I'm trying to prove DCT with Egorov theorem.
Let $E$ s.t $m(E)<\varepsilon$, $f_n(x)\to f(x)$ a.e. on $E$ and $|f_n|\leq \varphi\in L^1$ for all $n$.
Let $\varepsilon>0$. There is $A_\varepsilon\subset E$ s.t. $m(E-A_\varepsilon)<\varepsilon$ and $f_n\to f$ uniformly. Then $$\int_E |f_n-f|=\int_{A_\varepsilon}|f_n-f|+\int_{ A_\varepsilon}|f_n-f|\leq \int_{A_\varepsilon}|f_n-f|+\int_{A_\varepsilon^c\cap E}|\varphi+f|.$$ I guess that $\int_{A_\varepsilon\cap E}|\varphi+f|<\varepsilon,$ but I don't see how to prove it.
Hint
Let $\varepsilon>0$. You have to use the fact that since $|f+\varphi|$ is integrable, there is $\delta>0$ s.t. $$\int_K |f+g|<\varepsilon,$$ whenever $m(K)<\delta$ with $K\subset E$. Then, take $A_\delta\subset E$ s.t. $m(A_\delta)<\delta$ and continue the proof as you did.