[donthireddy]

Thanks to eakarahan for asking, and to Prof. Lin for designing a great exercise.

I think I have managed to complete the proof. Most of the steps pushed my math skills to the limit. Took 2-3 hours spread over a couple of days.
The first one made me think harder about what a probability distribution is.
A fair bit of algebraic manipulation in the other steps, some optimization in step 4. I needed to think of using the Taylor expansion for one of the steps. Overall, it was excellent exercise for the brain.