How could one come up with the proof of Azuma inequality? I'm looking for intuition for the proof. I'm reading the proof provided here: http://willperkins.org/6221/slides/azuma.pdf
I have a martingle $X_1,X_2...X_m$with small differences, why should I think of bounding $E[e^{a{X_m}}]$? Why not some other function? After making this steps I think I understand why we do the rest of the stuff, but this intial step...
Thanks