Sometimes I construct a solution, than I prove the solution is the best by proving it's not possible to get a better solution by modifying my solution by one step. (wow, a lot "by"s, forgive me XD)
For example The tutorial also use this method to prove.
The proof BASES ON the solution.
What if we modify more? From the second step on we cannot prove it anymore because the state before is not the solution.
That is to say :
We have solution1. We can change it to solution2,3 by one step and we know they are not better. We can change solution2 to solution4,5, and change solution4 to solution6,7,8, ans so on. The proof base on solution1, so, wo cannot know wether solution4 is not better than solution2, which means we cannot say that solution4 is not better than solution1.
We can only say that solutions near solution1 is not better than it. So maybe it's loose to say solution1 is the best.