From Van H. Vu’s “Concentration of Non-Lipschitz Functions and Applications,” Random Structures and Algorithms 20 : 262–316, 2002 :
Finding this hypothesis was actually the hardest part of our work, requiring some insight and a numerous number of attempts. As the reader will see after reading Section 5, a proper induction hypothesis not only allows us to derive a powerful result, but also reduces the proof to a rather routine verification of a few conditions.
I have spent a long time reading through difficult proofs with numerous lemmata, wondering why it had to be so complicated. Some things have to be proved by brute force. For others, just phrasing the problem in the right way can make the proof seem trivial. Some might say “well, I could have done that,” but the more accurate response is “I wish I had thought to do it that way.”