The theorem and its proof are given below:
My questions are:
1-It is not clear for me how the third line from below and the 2 lines above it proof that $\{f_{n}\}$ converges uniformly to $f$ on $A$, could anyone explain this for me, please?
2-Secondly, it is not clear for me from the comment after the proof, why Egoroff's theorem also holds if the limit function is finite a.e., could anyone explain this for me, please?